# Patterns in Permutations

This challenge is based partly on this MSE question and exists to extend some OEIS sequences, and create others. If I extend or create sequences based on this challenge, I'll link to this challenge so that folks can see where the values come from. If you'd like direct credit, include your first and last name so I can cite you on the entry.

These are now tables in the OEIS as A342840 (1342), A342860 (2413), A342861 (1324), A342862 (2143), A342863 (1243), A342864 (1432), and A342865 (1234).

## Definition

Let $$\\sigma \in S_n\$$ be a permutation on $$\n\$$ letters, and let $$\\tau \in S_m\$$ be a permution on $$\m \leq n\$$ letters, both read as words: $$\sigma = \sigma_1\sigma_2 \dots \sigma_n \text{ and } \tau = \tau_1\tau_2 \dots \tau_m$$ The permutation statistic $$\\operatorname{pat}_\tau(\sigma)\$$ counts the number of length-$$\m\$$ subsequences of $$\\sigma\$$ that are in the same relative order as $$\\tau\$$.

## Challenge

This challenge will have you take a permutation $$\\tau \in S_m\$$ and an integer $$\n \geq m\$$ and return a (0-indexed) list where position $$\i\$$ has the number of permutations $$\\sigma \in S_n\$$ such that $$\\operatorname{pat}_{\tau}(\sigma) = i\$$. If done correctly, the list will sum to $$\n!\$$.

## Scoring

I will run your code on each of the following for one minute each for increasing values of $$\n\$$:

• the only pattern in $$\S_1\$$: $$\\tau = 1\$$;
• essentially the only pattern in $$\S_2\$$: $$\\tau = 12\$$;
• the two essentially different patterns in $$\S_3\$$: $$\\tau \in \{123, 132\}\$$; and
• the seven essentially distinct patterns of $$\S_4\$$: $$\ \tau \in \{1234, 1243, 1324, 1342, 1432, 2143, 2413\}\$$.

Your score will be the largest value of $$\n\$$ that can be computed within a minute for every pattern. For example, if you can compute up to

• $$\n = 1000\$$ for $$\\tau = 123\$$ within a minute,
• $$\n = 10\$$ for $$\\tau = 1342\$$ within a minute, and
• $$\n = 11\$$ for $$\\tau = 2413\$$ within a minute,

then your score would be $$\10\$$.

### Examples

Suppose that $$\\sigma = 7532146\$$ and $$\\tau = 4213\$$. Then $$\operatorname{pat}_{4213}(7532146) = 13$$ because $$\\sigma = 7532146\$$ has thirteen subsequences that are in the same relative order as $$\\tau = 4213\$$, $$\7536\$$, $$\7526\$$, $$\7516\$$, $$\7546\$$, $$\7324\$$, $$\7326\$$, $$\7314\$$, $$\7316\$$, $$\7214\$$, $$\7216\$$, $$\5324\$$, $$\5314\$$, and $$\5214\$$.

### Test Data

 n | tau   | target output
---+-------+--------------------------------------------------------------
6 |  1234 | [513, 102, 63, 10,  6, 12, 8, 0, 0,      5, 0, 0, 0, 0, 0, 1]
5 |   123 | [ 42,  27, 24,  7,  9,  6, 0, 4, 0,      0, 1]
9 |     1 | [  0,   0,  0,  0,  0,  0, 0, 0, 0, 362880]
5 |   132 | [ 42,  21, 23, 14, 12,  5, 3]
6 |  4213 | [512,  77, 69, 30, 21,  5, 6]
4 |    12 | [  1,   3,  5,  6,  5,  3, 1]
6 | 53214 | [694,  18,  7,  0,  1]


Note: For $$\\tau = 12\$$, the target output should be the $$\n\$$-th row of OEIS sequence A008302. For $$\\tau = 1\$$, the target output should be $$\n\$$ $$\0\$$s followed by $$\n!\$$.

• I didn't quite understand what "subsequences that are in the same relative order as tau" means here. Does it mean the subsequence contains tau as its subsequence? Or any two letters in the subsequence will either 1. one or two of them not in tau, or 2. both them are in tau, and the order of these two letters is as same as in both subsequence or tau?
– tsh
Commented Mar 25, 2021 at 8:05
• What are $S_n$ and $S_m$? Should I understand that they are two sets, and whenever $m\le n$, we have $S_m \subseteq S_n$?
– tsh
Commented Mar 25, 2021 at 8:07
• $S_n$ is just the set of a all permutations of $\{1,2,3,\dots,n\}$. Strictly speaking, $S_m$ is not a subset of this unless $m=n$. But I suspect I may be misunderstanding what you’re getting at. Commented Mar 25, 2021 at 17:04
• @tsh, "Subsequences that are in the same relative order as tau" means that if $\tau$ is $4123$, then we're looking for length-4 subsequences where the first number is the biggest, the second is the smallest, the third is the second-smallest, and the last number is the second-largest. Commented Mar 25, 2021 at 20:05
• $1342$ and $1423$ are not essentially distinct, since they’re inverses. Commented Mar 28, 2021 at 4:28

# Rust, score ≈ 13

This enumerates permutations σ recursively, sharing as much computation as possible at branches of the search tree, so that only constant work is needed at each leaf.

Times in seconds measured on my Ryzen 7 1800X (8 cores/16 threads):

τ   n = 10    11    12     13      14       15
──────────────────────────────────────────────
1     0.00  0.02  0.25   2.93   38.72   660.86
12    0.00  0.04  0.47   5.89   79.08  1268.18
123   0.00  0.06  0.67   8.47  115.61  1836.36
132   0.01  0.10  1.18  15.50  218.39  3443.56
1234  0.00  0.05  0.59   7.65  104.88  1643.29
1243  0.00  0.06  0.81  10.72  152.60  2417.75
1324  0.01  0.09  1.10  14.52  208.59  3280.18
1342  0.01  0.10  1.21  16.59  242.01  3871.64
1432  0.01  0.09  1.19  16.21  237.93  3808.25
2143  0.00  0.06  0.81  10.72  151.09  2409.76
2413  0.01  0.09  1.20  16.07  232.74  3717.70
──────────────────────────────────────────────
max   0.01  0.10  1.21  16.59  242.01  3871.64


Build and run with:

cargo build --release
target/release/permutations 11 '1 3 4 2'


Or try it online! (But note, the TIO version lacks parallelism and many other optimizations, so it’s much slower, due to being unable to use external crates.)

All outputs for n = 0…15.

Cargo.toml

[package]
name = "permutations"
version = "0.1.0"
authors = ["Anders Kaseorg <[email protected]>"]
edition = "2018"

[dependencies]
rayon = "1.5.0"
typed-arena = "2.0.1"

[profile.release]
lto = "y"
panic = "abort"


src/main.rs

use rayon::prelude::*;
use std::cell::Cell;
use std::env;
use std::iter::Zip;
use std::slice;
use typed_arena::Arena;

enum UpdateIterator<'a> {
Leave(u16, slice::Iter<'a, (u16, u16)>),
Take(
u16,
u16,
Zip<slice::Iter<'a, (u16, u16)>, slice::Iter<'a, u16>>,
),
}

impl Iterator for UpdateIterator<'_> {
type Item = (u16, u16);

fn next(&mut self) -> Option<(u16, u16)> {
match *self {
UpdateIterator::Leave(i, ref mut bounds) => bounds
.next()
.map(|&(low, high)| (low + (low > i) as u16, high + (high >= i) as u16)),
UpdateIterator::Take(i, tau0, ref mut zip) => {
zip.next().map(|(&(low, high), &tau1)| {
if tau0 < tau1 {
(low.max(i) + 1, high + (high >= i) as u16)
} else {
(low + (low > i) as u16, high.min(i))
}
})
}
}
}

fn size_hint(&self) -> (usize, Option<usize>) {
match self {
UpdateIterator::Leave(_, bounds) => bounds.size_hint(),
UpdateIterator::Take(_, _, zip) => zip.size_hint(),
}
}
}

fn update(
tau: &[u16],
remaining: usize,
states: &[(&[(u16, u16)], isize)],
i: u16,
mut callback: impl FnMut(UpdateIterator, isize),
) {
for &(bounds, count) in states {
if bounds.len() < remaining {
callback(UpdateIterator::Leave(i, bounds.iter()), count);
}

if let Some((&(low0, high0), bounds1)) = bounds.split_first() {
if low0 <= i && i <= high0 {
callback(
UpdateIterator::Take(
i,
tau[tau.len() - bounds.len()],
bounds1.iter().zip(&tau[tau.len() - bounds.len() + 1..]),
),
count,
);
}
}
}
}

fn search_step(
tau: &[u16],
remaining: usize,
states: &[(&[(u16, u16)], isize)],
arena_len: usize,
i: u16,
search: impl FnOnce(&[(&[(u16, u16)], isize)], usize),
) {
let arena = Arena::with_capacity(2 * arena_len + 1 - states.len());
let mut new_states = Vec::with_capacity(states.len() * 2);
update(tau, remaining, states, i, |bounds, count| {
new_states.push((&*arena.alloc_extend(bounds), count))
});
new_states.sort_by_key(|&(bounds, _)| bounds);
new_states.dedup_by(|(bounds0, count0), (bounds1, count1)| {
bounds0 == bounds1 && {
*count1 += *count0;
true
}
});
search(&new_states, arena.len());
}

fn search(
n: usize,
tau: &[u16],
k: usize,
states: &[(&[(u16, u16)], isize)],
arena_len: usize,
output: &[Cell<u64>],
) {
if n - k == 1 {
let mut deltas = vec![0; n + 1];
let mut total = 0;
for &(bounds, count) in states {
match bounds {
[] => total += count,
&[(low, high)] => {
deltas[low as usize] += count;
deltas[high as usize + 1] -= count;
}
_ => {}
}
}
for delta in &deltas[..n] {
total += delta;
output[total as usize].set(output[total as usize].get() + 1);
}
} else if n - k == 2 {
for i in 0..=k as u16 {
let mut deltas = vec![0; n + 1];
let mut total = 0;
update(tau, n - k, states, i, |mut bounds, count| {
if let Some((low, high)) = bounds.next() {
deltas[low as usize] += count;
deltas[high as usize + 1] -= count;
debug_assert!(bounds.next().is_none());
} else {
total += count;
}
});
for delta in &deltas[..n] {
total += delta;
output[total as usize].set(output[total as usize].get() + 1);
}
}
} else {
for i in 0..=k as u16 {
search_step(tau, n - k, states, arena_len, i, |states, arena_len| {
search(n, tau, k + 1, states, arena_len, output)
})
}
}
}

fn par_search(
n: usize,
tau: &[u16],
k: usize,
states: &[(&[(u16, u16)], isize)],
arena_len: usize,
comb: usize,
) {
if n - k <= 4 {
search(
n,
tau,
k,
states,
arena_len,
outputs.get_or(|| vec![Cell::new(0); comb + 1]),
);
} else {
(0..=k as u16).into_par_iter().for_each(|i| {
search_step(tau, n - k, states, arena_len, i, |states, arena_len| {
par_search(n, tau, k + 1, states, arena_len, comb, outputs)
})
});
}
}

fn counts(n: usize, tau: &[u16]) -> Vec<u64> {
if n > tau.len() {
let mut comb = 1;
for i in 0..tau.len() {
comb = comb * (n - i) / (i + 1);
}
par_search(
n,
&tau,
0,
&[(&vec![(0, 0); tau.len()], 1)],
tau.len(),
comb,
&outputs,
);
let mut output = vec![0; comb + 1];
for (x, y) in output.iter_mut().zip(thread_output) {
*x += y.get();
}
}
output.truncate(
output
.iter()
.rposition(|&count| count != 0)
.map_or(0, |i| i + 1),
);
output
} else if n == tau.len() {
vec![(1..=n as u64).product::<u64>() - 1, 1]
} else {
vec![(1..=n as u64).product()]
}
}

fn main() {
let args: Vec<String> = env::args().collect();
if let [_, n, tau] = &*args {
let n: usize = n.parse().unwrap();
let tau: Vec<u16> = tau.split_whitespace().map(|n| n.parse().unwrap()).collect();
for count in counts(n, &tau) {
println!("{}", count);
}
} else {
panic!("usage: permutations n 'τ₁ τ₂ … τₘ'")
}
}


# JavaScript (Node.js), Score: 10 (estimated)

For now, this is a very simple implementation to make sure I understood the challenge correctly. The score is probably just 10, which is quite bad.

function count(a, b, c = [], p = 0, i = 0) {
return (
p == b.length ? 1 :
i == a.length ? 0 :
(
b.every((v, k) => k >= p || Math.sign(b[p] - v) == Math.sign(a[i] - c[k])) &&
count(a, b, [...c, a[i]], p + 1, i + 1)
) +
count(a, b, c, p, i + 1)
);
}

function perm(a) {
let r = [];

(function P(a, p = []) {
if(a.length) {
a.forEach((v, i) => P(a.filter(_ => i--), [...p, v]));
}
else {
r.push(p);
}
})(a);

return r;
}

function f(n, t) {
let list = [];

perm([...Array(n)].map((_, i) => i + 1)).forEach(p => {
let x = count(p, t);

if(!list[x]) {
for(let i = 0; i <= x; i++) list[i] |= 0;
}
list[x]++;
});
return list;
}


Try it online!

# C++ (clang), score 11

#include "patterns_in_permutations.hpp"

int main(int argc, char *argv[])
{
unsigned int running_time = 60;
unsigned int n = 4;
std::unordered_map<size_t, std::vector<std::string>> all_gens;
for (const auto finish = start + std::chrono::seconds{running_time}; n < 12 && std::chrono::steady_clock::now() < finish; ++n)
{
Perms p(n);
const std::vector<std::string> gens = p.Generate();
all_gens[n] = gens;
Runner1(n);
}
const std::chrono::duration<double> duration = std::chrono::steady_clock::now() - start;
std::cout << "It took " << duration.count() << " seconds.\n";
Runner(Runner2, running_time, 4, "12", all_gens);
Runner(Runner3, running_time, 4, "123", all_gens);
Runner(Runner3, running_time, 4, "132", all_gens);
Runner(Runner4, running_time, 4, "1234", all_gens);
Runner(Runner4, running_time, 4, "1243", all_gens);
Runner(Runner4, running_time, 4, "1324", all_gens);
Runner(Runner4, running_time, 4, "1342", all_gens);
Runner(Runner4, running_time, 4, "1423", all_gens);
Runner(Runner4, running_time, 4, "1432", all_gens);
Runner(Runner4, running_time, 4, "2143", all_gens);
Runner(Runner4, running_time, 4, "2413", all_gens);

return 0;
}


### patterns_in_permutations.hpp:

#include <iostream>
#include <string>
#include <vector>
#include <functional>
#include <algorithm>
#include <iterator>
#include <chrono>
#include <unordered_map>

class Perms
{
public:
Perms(size_t n)
{
char next = '1';
for (int i = 0; i < n; ++i) {
base_+= next;
++next;
}
}

std::vector<std::string> Generate()
{
std::string current = base_;
do {
perms_.push_back(current);
} while (std::next_permutation(current.begin(), current.end()));
return perms_;
}
private:
std::vector<std::string> perms_;
std::string base_;
};

class Pattern2
{
public:
Pattern2()
{}
unsigned int CountMatches(const std::string& word) const
{
unsigned int matches = 0;
size_t size = word.size();
for (int i = 0; i < size - 1; ++i)
for (int j = i + 1; j < size; ++j)
matches += word[i] < word[j];
return matches;
}
};

class Pattern3
{
public:
Pattern3(const std::string& word) : pattern_{GenPattern(word[0], word[1], word[2])}
{}
unsigned int GenPattern(char a, char b, char c) const
{
unsigned int pattern = (a>b) | ((a>c)<<1) | ((b>c)<<2);
return pattern;
}
bool Matches(char a, char b, char c) const
{
unsigned int pattern = GenPattern(a, b, c);
return pattern == pattern_;
}
unsigned int CountMatches(const std::string& word) const
{
unsigned int matches = 0;
size_t size = word.size();
for (int i = 0; i < size - 2; ++i)
for (int j = i + 1; j < size - 1; ++j)
for (int k = j + 1; k < size; ++k)
matches += Matches(word[i], word[j], word[k]);
return matches;
}
private:
unsigned int pattern_;
};

class Pattern4
{
public:
Pattern4(const std::string& word) : pattern_{GenPattern(word[0], word[1], word[2], word[3])}
{}
unsigned int GenPattern(char a, char b, char c, char d) const
{
unsigned int pattern = (a>b) | ((a>c)<<1) |  ((a>d)<<2) | ((b>c)<<3) | ((b>d)<<4) |  ((c>d)<<5);
return pattern;
}
bool Matches(char a, char b, char c, char d) const
{
unsigned int pattern = GenPattern(a, b, c, d);
return pattern == pattern_;
}
unsigned int CountMatches(const std::string& word) const
{
unsigned int matches = 0;
size_t size = word.size();
for (int i = 0; i < size - 3; ++i)
for (int j = i + 1; j < size - 2; ++j)
for (int k = j + 1; k < size - 1; ++k)
for (int l = k + 1; l < size; ++l)
matches += Matches(word[i], word[j], word[k], word[l]);
return matches;
}
private:
unsigned int pattern_;
};

void Print(unsigned int n, const std::string& tau, const std::vector<unsigned long>& counts)
{
std::cout << "n is: " << n << ", tau is: " << tau << ", and we have: [";
int i = 0;
for (auto c : counts) {
std::cout << (i ? ", " : "") << c;
++i;
}
std::cout << "]\n";
}

void Runner1(unsigned int n)
{
unsigned int num_counts = n + 1;
std::vector<unsigned long> counts(num_counts, 0);
counts[n] = 1;
for (int i = 2; i <= n; ++i) {
counts[n] *= i;
}
Print(n, "1", counts);
}

void Runner2(const std::string&, unsigned int n, const std::vector<std::string>& gens)
{
Pattern2 pat;
unsigned int num_counts = n*n;
std::vector<unsigned long> counts(num_counts, 0);
for (const std::string& sigma : gens) {
auto matches = pat.CountMatches(sigma);
++counts[matches];
}
while (counts.back() == 0) {
counts.pop_back();
}
Print(n, "12", counts);
}

void Counter3(std::vector<unsigned long>& counts,
const std::vector<std::string>& gens,
const std::vector<std::string>::const_iterator& start,
const std::vector<std::string>::const_iterator& end,
const Pattern3& pat)
{
for (auto iter = start; iter < end; ++iter) {
auto matches = pat.CountMatches(*iter);
++counts[matches];
}
}

void Runner3(const std::string& tau, unsigned int n, const std::vector<std::string>& gens)
{
Pattern3 pat{tau};
unsigned int num_counts = n * n * (n - 1) * (n - 2);

size_t block_size = gens.size() / 4;
std::vector<unsigned long> counts0(num_counts, 0);
auto start0 = gens.begin();
auto end0 = start0;
std::vector<unsigned long> counts1(num_counts, 0);
auto start1 = end0;
auto end1 = start1;
std::vector<unsigned long> counts2(num_counts, 0);
auto start2 = end1;
auto end2 = start2;
std::vector<unsigned long> counts3(num_counts, 0);
auto start3 = end2;
auto end3 = gens.end();
std::thread t0(Counter3, std::ref(counts0), std::cref(gens), std::cref(start0), std::cref(end0), std::cref(pat));
std::thread t1(Counter3, std::ref(counts1), std::cref(gens), std::cref(start1), std::cref(end1), std::cref(pat));
std::thread t2(Counter3, std::ref(counts2), std::cref(gens), std::cref(start2), std::cref(end2), std::cref(pat));
std::thread t3(Counter3, std::ref(counts3), std::cref(gens), std::cref(start3), std::cref(end3), std::cref(pat));
t0.join();
t1.join();
t2.join();
t3.join();
std::vector<unsigned long> counts(num_counts, 0);
for (int i = 0; i < num_counts; ++i) {
counts[i] = counts0[i] + counts1[i] + counts2[i] + counts3[i];
}

while (counts.back() == 0) {
counts.pop_back();
}
Print(n, tau, counts);
}

void Counter4(std::vector<unsigned long>& counts,
const std::vector<std::string>& gens,
const std::vector<std::string>::const_iterator& start,
const std::vector<std::string>::const_iterator& end,
const Pattern4& pat)
{
for (auto iter = start; iter < end; ++iter) {
auto matches = pat.CountMatches(*iter);
++counts[matches];
}
}

void Runner4(const std::string& tau, unsigned int n, const std::vector<std::string>& gens)
{
Pattern4 pat{tau};
unsigned int num_counts = n * n * (n - 1) * (n - 2) * (n - 3);

size_t block_size = gens.size() / 4;
std::vector<unsigned long> counts0(num_counts, 0);
auto start0 = gens.begin();
auto end0 = start0;
std::vector<unsigned long> counts1(num_counts, 0);
auto start1 = end0;
auto end1 = start1;
std::vector<unsigned long> counts2(num_counts, 0);
auto start2 = end1;
auto end2 = start2;
std::vector<unsigned long> counts3(num_counts, 0);
auto start3 = end2;
auto end3 = gens.end();
std::thread t0(Counter4, std::ref(counts0), std::cref(gens), std::cref(start0), std::cref(end0), std::cref(pat));
std::thread t1(Counter4, std::ref(counts1), std::cref(gens), std::cref(start1), std::cref(end1), std::cref(pat));
std::thread t2(Counter4, std::ref(counts2), std::cref(gens), std::cref(start2), std::cref(end2), std::cref(pat));
std::thread t3(Counter4, std::ref(counts3), std::cref(gens), std::cref(start3), std::cref(end3), std::cref(pat));
t0.join();
t1.join();
t2.join();
t3.join();
std::vector<unsigned long> counts(num_counts, 0);
for (int i = 0; i < num_counts; ++i) {
counts[i] = counts0[i] + counts1[i] + counts2[i] + counts3[i];
}

while (counts.back() == 0) {
counts.pop_back();
}
Print(n, tau, counts);
}

template<typename Func>
void Runner(Func func, unsigned int running_time, unsigned int n, const std::string& tau, std::unordered_map<size_t, std::vector<std::string>>& all_gens)
{
for (const auto finish = start + std::chrono::seconds{running_time}; n < 12 && std::chrono::steady_clock::now() < finish; ++n)
{
const std::vector<std::string> gens = all_gens[n];
func(tau, n, gens);
}
const std::chrono::duration<double> duration = std::chrono::steady_clock::now() - start;
std::cout << "It took " << duration.count() << " seconds.\n";
}


### Outputs:

n is: 4, tau is: 1, and we have: [0, 0, 0, 0, 24]
n is: 5, tau is: 1, and we have: [0, 0, 0, 0, 0, 120]
n is: 6, tau is: 1, and we have: [0, 0, 0, 0, 0, 0, 720]
n is: 7, tau is: 1, and we have: [0, 0, 0, 0, 0, 0, 0, 5040]
n is: 8, tau is: 1, and we have: [0, 0, 0, 0, 0, 0, 0, 0, 40320]
n is: 9, tau is: 1, and we have: [0, 0, 0, 0, 0, 0, 0, 0, 0, 362880]
n is: 10, tau is: 1, and we have: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800]
n is: 11, tau is: 1, and we have: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 39916800]
It took 2.29098 seconds.
n is: 4, tau is: 12, and we have: [1, 3, 5, 6, 5, 3, 1]
n is: 5, tau is: 12, and we have: [1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1]
n is: 6, tau is: 12, and we have: [1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1]
n is: 7, tau is: 12, and we have: [1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1]
n is: 8, tau is: 12, and we have: [1, 7, 27, 76, 174, 343, 602, 961, 1415, 1940, 2493, 3017, 3450, 3736, 3836, 3736, 3450, 3017, 2493, 1940, 1415, 961, 602, 343, 174, 76, 27, 7, 1]
n is: 9, tau is: 12, and we have: [1, 8, 35, 111, 285, 628, 1230, 2191, 3606, 5545, 8031, 11021, 14395, 17957, 21450, 24584, 27073, 28675, 29228, 28675, 27073, 24584, 21450, 17957, 14395, 11021, 8031, 5545, 3606, 2191, 1230, 628, 285, 111, 35, 8, 1]
n is: 10, tau is: 12, and we have: [1, 9, 44, 155, 440, 1068, 2298, 4489, 8095, 13640, 21670, 32683, 47043, 64889, 86054, 110010, 135853, 162337, 187959, 211089, 230131, 243694, 250749, 250749, 243694, 230131, 211089, 187959, 162337, 135853, 110010, 86054, 64889, 47043, 32683, 21670, 13640, 8095, 4489, 2298, 1068, 440, 155, 44, 9, 1]
n is: 11, tau is: 12, and we have: [1, 10, 54, 209, 649, 1717, 4015, 8504, 16599, 30239, 51909, 84591, 131625, 196470, 282369, 391939, 526724, 686763, 870233, 1073227, 1289718, 1511742, 1729808, 1933514, 2112319, 2256396, 2357475, 2409581, 2409581, 2357475, 2256396, 2112319, 1933514, 1729808, 1511742, 1289718, 1073227, 870233, 686763, 526724, 391939, 282369, 196470, 131625, 84591, 51909, 30239, 16599, 8504, 4015, 1717, 649, 209, 54, 10, 1]
It took 2.13947 seconds.
n is: 4, tau is: 123, and we have: [14, 6, 3, 0, 1]
n is: 5, tau is: 123, and we have: [42, 27, 24, 7, 9, 6, 0, 4, 0, 0, 1]
n is: 6, tau is: 123, and we have: [132, 110, 133, 70, 74, 54, 37, 32, 24, 12, 16, 6, 6, 8, 0, 0, 5, 0, 0, 0, 1]
n is: 7, tau is: 123, and we have: [429, 429, 635, 461, 507, 395, 387, 320, 260, 232, 191, 162, 104, 130, 100, 24, 74, 62, 18, 32, 10, 30, 13, 8, 0, 10, 10, 0, 0, 0, 6, 0, 0, 0, 0, 1]
n is: 8, tau is: 123, and we have: [1430, 1638, 2807, 2528, 3008, 2570, 2864, 2544, 2389, 2182, 2077, 1818, 1580, 1456, 1494, 886, 1047, 1004, 682, 656, 546, 466, 537, 288, 228, 324, 252, 156, 115, 138, 154, 66, 58, 60, 68, 38, 47, 0, 18, 40, 16, 10, 0, 0, 15, 12, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 1]
n is: 9, tau is: 123, and we have: [4862, 6188, 11864, 12525, 16151, 15203, 18179, 17357, 18096, 17333, 17505, 16605, 15847, 15068, 15049, 12630, 12472, 12101, 10837, 9588, 8935, 8225, 8089, 6836, 5405, 6072, 5158, 4541, 3901, 3462, 3412, 2976, 2524, 2151, 1887, 1995, 1583, 1312, 1064, 1190, 850, 834, 823, 508, 488, 420, 458, 427, 282, 186, 166, 234, 148, 248, 44, 80, 57, 66, 110, 58, 50, 0, 10, 20, 60, 19, 12, 0, 0, 0, 21, 14, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 1]
n is: 10, tau is: 123, and we have: [16796, 23256, 48756, 58258, 80889, 83382, 105082, 107194, 120197, 121630, 129449, 128712, 132579, 130310, 133945, 124572, 127049, 121602, 121330, 112240, 109115, 103134, 102875, 95256, 86372, 84020, 82579, 73540, 69645, 64590, 61259, 56574, 53742, 47734, 44615, 41368, 39552, 35538, 31990, 29182, 27361, 25088, 23871, 20574, 18612, 16776, 15679, 14582, 14304, 11058, 9812, 9188, 8974, 9012, 6788, 5824, 5451, 4708, 5058, 4682, 3486, 3004, 2795, 2206, 2678, 2434, 1977, 1256, 1192, 1134, 1480, 1130, 986, 606, 401, 608, 603, 700, 378, 312, 192, 156, 264, 296, 326, 60, 107, 64, 15, 120, 172, 72, 59, 0, 12, 0, 35, 84, 22, 14, 0, 0, 0, 0, 28, 16, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 1]
n is: 11, tau is: 123, and we have: [58786, 87210, 196707, 259787, 385387, 431167, 568809, 616449, 730182, 778432, 868607, 901457, 975820, 1000220, 1059437, 1051194, 1099257, 1092656, 1121794, 1101327, 1106169, 1079286, 1094214, 1065888, 1031696, 1005574, 1002920, 963317, 925079, 897540, 867777, 833036, 807892, 761461, 735291, 694925, 665451, 644967, 601738, 567469, 536582, 514598, 487819, 457925, 428537, 402525, 377705, 354636, 343753, 315180, 288746, 273107, 250591, 248644, 223830, 206056, 191347, 177756, 167509, 159312, 140646, 133790, 124559, 111283, 105226, 98752, 89678, 84846, 74486, 68371, 65623, 60868, 54635, 50256, 43978, 42730, 39349, 35811, 32045, 28528, 27264, 25732, 21425, 20715, 19063, 16286, 14829, 15964, 11766, 11168, 11210, 8690, 8278, 8326, 7356, 6657, 5607, 4362, 4878, 3660, 3955, 4146, 3456, 2574, 1787, 1862, 1688, 2283, 1806, 1756, 1310, 874, 398, 746, 867, 1089, 850, 534, 508, 166, 150, 166, 533, 392, 462, 64, 148, 66, 0, 35, 222, 224, 98, 68, 0, 14, 0, 0, 56, 112, 25, 16, 0, 0, 0, 0, 0, 36, 18, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 1]
It took 4.41592 seconds.
n is: 4, tau is: 132, and we have: [14, 5, 4, 1]
n is: 5, tau is: 132, and we have: [42, 21, 23, 14, 12, 5, 3]
n is: 6, tau is: 132, and we have: [132, 84, 107, 82, 96, 55, 64, 37, 29, 22, 10, 0, 2]
n is: 7, tau is: 132, and we have: [429, 330, 464, 410, 526, 394, 475, 365, 360, 298, 281, 175, 206, 126, 93, 55, 23, 14, 13, 1, 2]
n is: 8, tau is: 132, and we have: [1430, 1287, 1950, 1918, 2593, 2225, 2858, 2489, 2682, 2401, 2620, 2088, 2321, 1853, 1770, 1576, 1417, 1152, 1048, 730, 647, 397, 322, 169, 162, 109, 41, 37, 20, 0, 7, 1]
n is: 9, tau is: 132, and we have: [4862, 5005, 8063, 8657, 12165, 11539, 15174, 14772, 16627, 16066, 18248, 16413, 18449, 16681, 17104, 16300, 16525, 14486, 14891, 12878, 12952, 11213, 10816, 8969, 8484, 7136, 6163, 4914, 4110, 3094, 2722, 1937, 1611, 1181, 950, 509, 510, 311, 107, 141, 85, 11, 37, 6, 0, 5, 1]
n is: 10, tau is: 132, and we have: [16796, 19448, 33033, 38225, 55482, 57064, 76381, 79768, 94243, 96248, 112709, 109422, 124827, 121352, 130154, 129704, 137826, 130013, 137479, 129923, 134624, 128037, 129817, 119910, 120597, 112847, 109851, 101464, 98255, 88302, 86333, 75684, 71042, 62466, 57142, 47504, 43980, 36867, 31539, 26333, 23803, 18854, 16276, 12781, 10735, 8453, 6665, 5023, 4158, 2658, 2056, 1584, 1040, 681, 562, 271, 175, 159, 83, 26, 44, 11, 4, 6, 1]
n is: 11, tau is: 132, and we have: [58786, 75582, 134576, 166322, 248509, 273612, 372506, 411132, 501367, 540405, 645884, 666591, 771103, 792470, 873391, 904364, 988292, 984496, 1062388, 1058312, 1121118, 1117511, 1167522, 1140122, 1182129, 1158052, 1180398, 1150418, 1167814, 1120075, 1134523, 1080774, 1075062, 1022688, 1008723, 948246, 933609, 874695, 839814, 777886, 749612, 684351, 647464, 583310, 541746, 487788, 449094, 396602, 365025, 316887, 287377, 251343, 226338, 190755, 170845, 144866, 126540, 104648, 90914, 73269, 62680, 50919, 41128, 32514, 26314, 19986, 16651, 12085, 9197, 7079, 5386, 3647, 2884, 2099, 1284, 1016, 769, 358, 343, 150, 111, 82, 41, 6, 19, 6, 4, 1]
It took 4.17272 seconds.
n is: 4, tau is: 1234, and we have: [23, 1]
n is: 5, tau is: 1234, and we have: [103, 12, 4, 0, 0, 1]
n is: 6, tau is: 1234, and we have: [513, 102, 63, 10, 6, 12, 8, 0, 0, 5, 0, 0, 0, 0, 0, 1]
n is: 7, tau is: 1234, and we have: [2761, 770, 665, 196, 146, 116, 142, 46, 10, 72, 32, 24, 0, 13, 0, 12, 18, 0, 0, 10, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
n is: 8, tau is: 1234, and we have: [15767, 5545, 5982, 2477, 2148, 1204, 1782, 885, 503, 804, 573, 600, 199, 312, 112, 156, 333, 115, 96, 136, 142, 12, 0, 89, 24, 84, 44, 24, 10, 41, 0, 0, 40, 0, 0, 28, 8, 0, 0, 10, 0, 15, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
n is: 9, tau is: 1234, and we have: [94359, 39220, 49748, 25886, 25190, 13188, 19936, 11533, 9599, 9533, 7775, 8585, 5312, 5586, 3004, 3006, 4679, 2718, 2776, 2102, 3081, 1323, 640, 1586, 1253, 1590, 928, 998, 502, 948, 620, 220, 746, 567, 272, 400, 408, 242, 54, 440, 158, 302, 124, 157, 30, 364, 112, 12, 161, 94, 48, 28, 16, 60, 0, 152, 44, 94, 40, 46, 0, 10, 40, 0, 0, 50, 20, 0, 0, 0, 12, 68, 0, 0, 10, 19, 0, 0, 0, 0, 12, 0, 0, 0, 0, 21, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
n is: 10, tau is: 1234, and we have: [586590, 276144, 396642, 244233, 260505, 142550, 210663, 130920, 132620, 113954, 100201, 102866, 85736, 82752, 55779, 51594, 64086, 46271, 48938, 36084, 49273, 30289, 24894, 25486, 25212, 29569, 20598, 20286, 15216, 18642, 16001, 10686, 14354, 12342, 12164, 9042, 9190, 8688, 5268, 8640, 6875, 7195, 4618, 6043, 2831, 6714, 5228, 2594, 3816, 3726, 4140, 1548, 1916, 2116, 992, 3908, 1846, 2534, 1824, 2104, 1208, 994, 1422, 953, 870, 1320, 1210, 737, 372, 779, 302, 1426, 586, 828, 424, 942, 350, 68, 747, 263, 265, 299, 160, 56, 422, 594, 140, 406, 16, 108, 246, 340, 300, 8, 104, 161, 50, 216, 0, 0, 116, 240, 0, 80, 15, 142, 72, 24, 0, 56, 187, 0, 40, 0, 0, 158, 60, 0, 0, 0, 12, 60, 0, 0, 0, 59, 12, 8, 0, 35, 10, 0, 0, 0, 0, 84, 12, 0, 0, 0, 22, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
n is: 11, tau is: 1234, and we have: [3763290, 1948212, 3089010, 2167834, 2493489, 1476655, 2136586, 1396483, 1574270, 1308353, 1231914, 1168760, 1139142, 1065192, 848857, 772876, 854099, 688165, 725885, 568412, 709188, 510887, 500661, 428278, 433048, 460477, 396052, 360136, 307833, 337855, 307073, 252242, 275869, 230346, 254906, 212275, 202396, 177857, 160884, 176735, 155432, 160778, 126044, 139380, 102473, 129824, 119286, 100572, 96254, 84524, 108448, 70634, 67477, 66313, 48076, 79778, 60889, 67211, 49818, 62162, 47384, 37102, 46130, 38655, 38438, 36768, 36972, 33390, 24838, 27562, 19970, 35462, 23347, 26997, 19102, 26876, 21432, 16120, 19400, 16220, 17410, 12648, 12176, 8489, 12720, 19653, 10636, 11919, 8356, 11398, 6688, 11942, 11340, 5242, 8694, 7886, 5136, 8846, 5104, 4526, 3704, 8888, 3228, 6718, 3714, 5102, 2336, 3804, 4226, 2584, 5818, 2710, 2235, 2682, 2162, 3645, 3398, 3668, 908, 2298, 2028, 1972, 2084, 1264, 360, 2673, 2228, 610, 564, 1882, 780, 2404, 664, 540, 101, 2426, 1492, 1344, 136, 500, 1064, 620, 1060, 184, 552, 1179, 354, 250, 416, 538, 508, 644, 10, 32, 460, 857, 568, 192, 0, 330, 296, 668, 504, 12, 0, 618, 143, 180, 16, 40, 92, 388, 10, 225, 320, 134, 12, 20, 0, 60, 324, 138, 120, 60, 24, 342, 66, 120, 0, 40, 40, 24, 103, 0, 0, 0, 78, 0, 60, 210, 0, 0, 66, 0, 0, 140, 84, 0, 0, 0, 12, 22, 84, 0, 10, 0, 0, 68, 0, 0, 12, 0, 0, 0, 0, 56, 0, 0, 0, 0, 14, 0, 112, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
It took 10.9059 seconds.
n is: 4, tau is: 1243, and we have: [23, 1]
n is: 5, tau is: 1243, and we have: [103, 11, 4, 2]
n is: 6, tau is: 1243, and we have: [513, 88, 56, 32, 14, 7, 9, 0, 0, 1]
n is: 7, tau is: 1243, and we have: [2761, 638, 543, 341, 235, 138, 173, 51, 42, 47, 34, 6, 17, 4, 0, 7, 1, 0, 2]
n is: 8, tau is: 1243, and we have: [15767, 4478, 4600, 3119, 2658, 1710, 2180, 972, 975, 877, 771, 356, 542, 233, 184, 266, 157, 81, 130, 41, 60, 49, 16, 16, 37, 8, 9, 13, 3, 0, 10, 1, 0, 0, 0, 0, 1]
n is: 9, tau is: 1243, and we have: [94359, 31199, 36691, 26602, 25756, 17628, 22984, 12381, 13705, 11786, 11395, 6832, 9438, 5252, 4870, 5314, 4350, 2787, 3611, 1905, 2415, 2032, 1200, 1029, 1510, 905, 794, 680, 579, 409, 541, 242, 295, 275, 130, 137, 296, 56, 45, 120, 77, 28, 86, 17, 21, 45, 15, 4, 23, 0, 9, 5, 3, 0, 8, 1, 0, 1, 0, 0, 2]
n is: 10, tau is: 1243, and we have: [586590, 218033, 284370, 218957, 231390, 166338, 221429, 133959, 156652, 134354, 137682, 94181, 125521, 80373, 80587, 80257, 74696, 53683, 65521, 42356, 50366, 43226, 33854, 27707, 36211, 25734, 24108, 21261, 20671, 15427, 18681, 11391, 12545, 11648, 8921, 8324, 10657, 6082, 5355, 5853, 5991, 3719, 4870, 2942, 2636, 3405, 2395, 1732, 2600, 1439, 1457, 1328, 1346, 740, 1183, 759, 674, 733, 469, 303, 656, 329, 190, 329, 307, 158, 221, 109, 96, 128, 117, 63, 108, 31, 42, 51, 37, 10, 43, 6, 26, 11, 14, 2, 23, 6, 0, 1, 0, 0, 11, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1]
n is: 11, tau is: 1243, and we have: [3763290, 1535207, 2174352, 1767837, 1994176, 1496134, 2028316, 1333828, 1613196, 1400002, 1491191, 1107853, 1450538, 1015596, 1068935, 1031355, 1025935, 793317, 946788, 684288, 792446, 693662, 618559, 517310, 636195, 490334, 483795, 432288, 438545, 352040, 408434, 292745, 311630, 284867, 255834, 234898, 267252, 191952, 185679, 177865, 186232, 142166, 160978, 116730, 115187, 121503, 102725, 86464, 102757, 74906, 75723, 69379, 67843, 50755, 61223, 49497, 47371, 43658, 35241, 31189, 41500, 26903, 25412, 26919, 24336, 20775, 22053, 15432, 16000, 15455, 14645, 11527, 14570, 8759, 8812, 9220, 8078, 6370, 8098, 4763, 6182, 5400, 4197, 3545, 4861, 3079, 2496, 2699, 2605, 1985, 2805, 1700, 1612, 1613, 1265, 1053, 1446, 659, 934, 727, 891, 550, 726, 376, 451, 469, 370, 193, 320, 176, 252, 257, 222, 110, 151, 87, 87, 87, 78, 27, 141, 30, 28, 32, 20, 36, 53, 11, 2, 14, 24, 0, 10, 0, 0, 15, 8, 0, 1, 0, 9, 1, 0, 1, 0, 0, 1, 0, 0, 0, 2]
It took 11.0133 seconds.
n is: 4, tau is: 1324, and we have: [23, 1]
n is: 5, tau is: 1324, and we have: [103, 10, 6, 1]
n is: 6, tau is: 1324, and we have: [513, 75, 74, 26, 17, 9, 6]
n is: 7, tau is: 1324, and we have: [2762, 522, 645, 321, 290, 130, 166, 47, 54, 48, 41, 4, 8, 2]
n is: 8, tau is: 1324, and we have: [15793, 3579, 5023, 3058, 3232, 1527, 2228, 874, 1159, 893, 875, 340, 503, 281, 269, 207, 156, 112, 123, 21, 54, 2, 0, 6, 5]
n is: 9, tau is: 1324, and we have: [94776, 24670, 37549, 26174, 30409, 15966, 23762, 10965, 15598, 11639, 12070, 6487, 9633, 5690, 6056, 5021, 4579, 3366, 4128, 1991, 2734, 1503, 1488, 1127, 1432, 765, 933, 657, 496, 262, 425, 106, 154, 74, 92, 26, 54, 8, 0, 7, 4, 0, 4]
n is: 10, tau is: 1324, and we have: [591950, 172198, 277089, 213122, 264667, 154452, 228665, 119090, 171635, 130545, 140681, 87346, 129648, 80843, 89519, 77927, 77496, 57636, 72997, 44129, 56802, 38567, 40481, 30585, 39332, 26288, 29031, 23149, 22663, 15442, 20266, 11940, 13360, 10107, 10516, 7633, 9318, 6070, 5707, 4886, 5013, 2983, 3876, 2168, 2205, 1851, 1824, 950, 1214, 526, 685, 386, 485, 74, 391, 116, 92, 32, 42, 3, 65, 32, 0, 0, 0, 0, 4, 4, 1]
n is: 11, tau is: 1324, and we have: [3824112, 1219974, 2043416, 1693787, 2213548, 1420513, 2086877, 1205632, 1719334, 1345649, 1499284, 1018334, 1487469, 982487, 1111324, 994064, 1041082, 795125, 999180, 685365, 853857, 640737, 685976, 541423, 679946, 501230, 546144, 458471, 479249, 364168, 448498, 315978, 353005, 288836, 301293, 241579, 283104, 210799, 217702, 188745, 197027, 145802, 170042, 125405, 132888, 113900, 115130, 86529, 99081, 69574, 79658, 61304, 64454, 46900, 56090, 38556, 42314, 32258, 33958, 24748, 29923, 19324, 21834, 16521, 15886, 12154, 14597, 8298, 9837, 6772, 7068, 4758, 6183, 2874, 4226, 2660, 2398, 1474, 1902, 751, 1314, 798, 482, 400, 388, 208, 316, 74, 87, 148, 86, 10, 32, 38, 24, 0, 16, 4, 14, 0, 0, 6, 0, 0, 0, 0, 1]
It took 11.162 seconds.
n is: 4, tau is: 1342, and we have: [23, 1]
n is: 5, tau is: 1342, and we have: [103, 10, 6, 1]
n is: 6, tau is: 1342, and we have: [512, 77, 69, 30, 21, 5, 6]
n is: 7, tau is: 1342, and we have: [2740, 548, 598, 330, 335, 123, 174, 58, 58, 37, 26, 3, 9, 1]
n is: 8, tau is: 1342, and we have: [15485, 3799, 4686, 2970, 3411, 1676, 2338, 1040, 1317, 878, 777, 363, 608, 230, 252, 165, 133, 30, 93, 26, 31, 4, 1, 3, 4]
n is: 9, tau is: 1342, and we have: [91245, 26165, 35148, 24550, 30182, 17185, 24685, 12976, 16867, 12248, 12360, 7203, 11086, 5692, 6391, 5194, 5006, 2751, 3917, 2019, 2482, 1622, 1371, 812, 1233, 490, 495, 416, 360, 157, 282, 54, 78, 41, 29, 22, 49, 7, 4, 0, 6]
n is: 10, tau is: 1342, and we have: [555662, 180512, 258390, 194524, 249925, 157765, 228949, 137892, 178897, 136866, 147875, 97336, 144013, 86383, 97551, 83482, 87825, 57805, 75538, 48428, 59365, 44334, 43055, 30914, 40620, 25178, 26230, 21239, 21735, 14478, 18540, 10413, 11956, 8481, 8007, 5559, 7822, 3944, 3937, 2917, 3450, 1677, 2394, 1149, 1250, 1028, 811, 379, 802, 216, 322, 236, 249, 90, 189, 50, 65, 35, 16, 2, 36, 0, 12]
n is: 11, tau is: 1342, and we have: [3475090, 1251832, 1882813, 1506786, 1998264, 1364500, 1991958, 1318382, 1727153, 1382970, 1540425, 1108017, 1581056, 1057910, 1212103, 1062295, 1152892, 843320, 1060298, 761390, 916286, 740645, 749373, 594592, 742181, 536033, 568444, 485474, 512375, 392372, 469748, 334596, 371632, 300252, 299179, 240512, 288754, 200025, 206566, 171109, 188723, 132675, 152423, 106675, 115972, 97837, 90729, 66595, 84156, 53183, 58167, 45687, 46707, 32569, 38358, 24868, 27436, 20574, 19746, 13524, 18937, 10616, 10981, 8781, 9015, 5447, 7130, 4028, 4673, 3378, 3130, 1960, 3506, 1238, 1538, 1296, 1094, 619, 941, 297, 546, 423, 198, 120, 356, 56, 82, 62, 26, 6, 80, 2, 14, 8, 8, 0, 2]
It took 11.0463 seconds.
n is: 4, tau is: 1423, and we have: [23, 1]
n is: 5, tau is: 1423, and we have: [103, 10, 6, 1]
n is: 6, tau is: 1423, and we have: [512, 77, 69, 30, 21, 5, 6]
n is: 7, tau is: 1423, and we have: [2740, 548, 598, 330, 335, 123, 174, 58, 58, 37, 26, 3, 9, 1]
n is: 8, tau is: 1423, and we have: [15485, 3799, 4686, 2970, 3411, 1676, 2338, 1040, 1317, 878, 777, 363, 608, 230, 252, 165, 133, 30, 93, 26, 31, 4, 1, 3, 4]
n is: 9, tau is: 1423, and we have: [91245, 26165, 35148, 24550, 30182, 17185, 24685, 12976, 16867, 12248, 12360, 7203, 11086, 5692, 6391, 5194, 5006, 2751, 3917, 2019, 2482, 1622, 1371, 812, 1233, 490, 495, 416, 360, 157, 282, 54, 78, 41, 29, 22, 49, 7, 4, 0, 6]
n is: 10, tau is: 1423, and we have: [555662, 180512, 258390, 194524, 249925, 157765, 228949, 137892, 178897, 136866, 147875, 97336, 144013, 86383, 97551, 83482, 87825, 57805, 75538, 48428, 59365, 44334, 43055, 30914, 40620, 25178, 26230, 21239, 21735, 14478, 18540, 10413, 11956, 8481, 8007, 5559, 7822, 3944, 3937, 2917, 3450, 1677, 2394, 1149, 1250, 1028, 811, 379, 802, 216, 322, 236, 249, 90, 189, 50, 65, 35, 16, 2, 36, 0, 12]
n is: 11, tau is: 1423, and we have: [3475090, 1251832, 1882813, 1506786, 1998264, 1364500, 1991958, 1318382, 1727153, 1382970, 1540425, 1108017, 1581056, 1057910, 1212103, 1062295, 1152892, 843320, 1060298, 761390, 916286, 740645, 749373, 594592, 742181, 536033, 568444, 485474, 512375, 392372, 469748, 334596, 371632, 300252, 299179, 240512, 288754, 200025, 206566, 171109, 188723, 132675, 152423, 106675, 115972, 97837, 90729, 66595, 84156, 53183, 58167, 45687, 46707, 32569, 38358, 24868, 27436, 20574, 19746, 13524, 18937, 10616, 10981, 8781, 9015, 5447, 7130, 4028, 4673, 3378, 3130, 1960, 3506, 1238, 1538, 1296, 1094, 619, 941, 297, 546, 423, 198, 120, 356, 56, 82, 62, 26, 6, 80, 2, 14, 8, 8, 0, 2]
It took 10.8298 seconds.
n is: 4, tau is: 1432, and we have: [23, 1]
n is: 5, tau is: 1432, and we have: [103, 11, 5, 0, 1]
n is: 6, tau is: 1432, and we have: [513, 87, 68, 17, 18, 10, 0, 4, 2, 0, 1]
n is: 7, tau is: 1432, and we have: [2761, 625, 626, 268, 274, 138, 112, 58, 51, 44, 31, 9, 15, 8, 12, 0, 5, 0, 0, 0, 3]
n is: 8, tau is: 1432, and we have: [15767, 4378, 5038, 2781, 3060, 1697, 1817, 1036, 964, 773, 656, 450, 379, 320, 285, 148, 237, 97, 98, 55, 68, 61, 23, 30, 30, 13, 30, 0, 0, 0, 16, 0, 10, 0, 0, 1, 0, 0, 0, 0, 2]
n is: 9, tau is: 1432, and we have: [94359, 30671, 38541, 24731, 28881, 17943, 21193, 13040, 14245, 10607, 10156, 7596, 7574, 5938, 5647, 3722, 4904, 3131, 3256, 2205, 2372, 1729, 1572, 1423, 1130, 846, 1014, 634, 644, 316, 609, 295, 371, 190, 306, 105, 195, 82, 94, 182, 86, 32, 79, 0, 49, 18, 41, 8, 30, 0, 43, 0, 20, 0, 0, 4, 1, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2]
n is: 10, tau is: 1432, and we have: [586590, 216883, 289785, 205853, 251051, 170941, 211942, 142187, 163587, 128113, 128732, 100701, 109971, 85489, 85649, 67021, 75089, 56874, 60397, 45806, 49198, 40624, 37457, 32975, 31394, 26058, 26056, 20705, 20996, 14894, 17646, 12853, 13931, 10583, 10699, 8066, 8780, 6271, 6318, 6394, 5366, 3753, 4414, 2917, 3838, 2395, 3069, 1715, 2167, 1181, 1746, 1424, 1579, 880, 927, 841, 617, 569, 697, 293, 765, 139, 373, 268, 329, 241, 317, 85, 200, 108, 202, 94, 92, 16, 24, 78, 92, 19, 140, 0, 107, 5, 16, 10, 1, 4, 0, 0, 30, 0, 72, 4, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 2]
n is: 11, tau is: 1432, and we have: [3763290, 1552588, 2172387, 1663964, 2096207, 1535129, 1954751, 1417314, 1674215, 1367302, 1442060, 1166386, 1334171, 1059609, 1109249, 930634, 1012528, 817526, 883392, 712076, 770293, 657724, 659051, 567799, 581450, 505877, 505195, 435477, 442807, 363235, 391358, 314376, 330559, 279469, 279737, 233779, 247628, 202488, 198986, 181873, 178728, 145947, 151991, 122028, 133974, 106370, 109130, 88316, 95025, 72848, 77553, 66516, 69104, 54015, 56357, 46890, 45853, 39525, 41001, 29289, 37365, 24516, 28021, 21883, 24707, 20150, 20127, 13574, 17768, 13105, 14200, 11333, 12459, 8252, 8839, 8040, 8981, 5471, 8179, 4280, 6717, 4186, 4545, 3061, 4806, 3268, 2641, 2630, 2670, 1591, 3442, 2364, 2218, 905, 1571, 704, 1344, 916, 734, 877, 1257, 599, 1219, 446, 889, 470, 843, 210, 418, 136, 385, 83, 150, 172, 356, 50, 452, 274, 212, 42, 380, 36, 16, 34, 40, 62, 56, 34, 138, 0, 110, 0, 106, 20, 6, 84, 0, 0, 0, 0, 69, 1, 28, 0, 0, 0, 0, 12, 0, 0, 42, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8]
It took 10.8561 seconds.
n is: 4, tau is: 2143, and we have: [23, 1]
n is: 5, tau is: 2143, and we have: [103, 11, 4, 2]
n is: 6, tau is: 2143, and we have: [513, 88, 53, 33, 18, 8, 6, 0, 0, 1]
n is: 7, tau is: 2143, and we have: [2761, 642, 495, 340, 262, 160, 172, 65, 58, 39, 14, 6, 18, 0, 0, 6, 0, 0, 2]
n is: 8, tau is: 2143, and we have: [15767, 4567, 4099, 3007, 2692, 1832, 2171, 1152, 1291, 968, 728, 457, 566, 174, 176, 221, 129, 14, 122, 29, 38, 52, 8, 0, 32, 9, 0, 10, 0, 0, 8, 0, 0, 0, 0, 0, 1]
n is: 9, tau is: 2143, and we have: [94359, 32443, 32345, 25049, 24492, 17732, 21841, 13234, 15867, 12824, 11744, 8852, 10670, 5983, 6058, 5709, 4751, 2372, 3642, 1790, 2080, 1799, 1020, 719, 1508, 621, 456, 549, 490, 200, 498, 152, 170, 198, 66, 114, 189, 34, 8, 52, 70, 0, 58, 4, 0, 30, 0, 0, 22, 0, 6, 0, 0, 0, 8, 0, 0, 0, 0, 0, 2]
n is: 10, tau is: 2143, and we have: [586590, 232189, 250371, 203452, 211291, 160561, 201524, 133030, 162800, 136840, 134669, 109219, 134085, 90985, 97178, 91917, 87126, 60426, 74360, 50859, 55948, 46889, 37498, 28547, 39584, 23350, 21951, 20022, 19986, 12961, 17766, 9861, 11022, 9968, 6998, 6849, 9424, 4124, 3624, 4367, 4884, 2464, 3844, 1630, 1911, 2721, 1054, 795, 2217, 783, 873, 778, 600, 240, 748, 514, 486, 276, 130, 68, 522, 30, 58, 214, 121, 126, 116, 10, 16, 41, 64, 0, 140, 4, 12, 10, 0, 0, 32, 0, 18, 16, 0, 0, 12, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
n is: 11, tau is: 2143, and we have: [3763290, 1679295, 1926145, 1635315, 1776655, 1409304, 1787218, 1263567, 1554692, 1348551, 1376606, 1161259, 1427223, 1058226, 1158896, 1110992, 1120317, 883389, 1044049, 817100, 905437, 795435, 730734, 615053, 747506, 548342, 542930, 488947, 490209, 377724, 439366, 313966, 333400, 293175, 258751, 232438, 276324, 179719, 175730, 167202, 179041, 127377, 149490, 100038, 107522, 110872, 82676, 66882, 95691, 59304, 59918, 56481, 50440, 36032, 49024, 36793, 36822, 31347, 22820, 17667, 34580, 16068, 14327, 18082, 16211, 13114, 13538, 7564, 8832, 7829, 9394, 4766, 11001, 4054, 3146, 5248, 3522, 3338, 4014, 1790, 4034, 2533, 1382, 994, 3394, 1542, 348, 790, 1528, 530, 2034, 880, 420, 370, 192, 288, 1138, 280, 210, 458, 321, 110, 308, 48, 316, 116, 62, 0, 350, 16, 140, 32, 98, 0, 64, 20, 0, 112, 0, 4, 108, 0, 0, 0, 4, 0, 42, 0, 0, 0, 32, 0, 0, 0, 0, 8, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2]
It took 10.9596 seconds.
n is: 4, tau is: 2413, and we have: [23, 1]
n is: 5, tau is: 2413, and we have: [103, 9, 8]
n is: 6, tau is: 2413, and we have: [512, 62, 82, 34, 28, 2]
n is: 7, tau is: 2413, and we have: [2740, 402, 612, 384, 466, 94, 232, 42, 60, 8]
n is: 8, tau is: 2413, and we have: [15485, 2593, 4187, 3036, 4356, 1746, 3132, 1064, 1918, 909, 654, 333, 612, 144, 104, 22, 24, 1]
n is: 9, tau is: 2413, and we have: [91245, 16921, 28065, 21638, 33274, 17598, 31180, 12942, 24000, 14290, 15434, 7770, 15692, 5965, 6896, 3947, 5660, 2226, 3674, 1314, 1512, 516, 508, 204, 332, 37, 40]
n is: 10, tau is: 2413, and we have: [555662, 112196, 188514, 149946, 237128, 140954, 257686, 132874, 222776, 149894, 184050, 106012, 211448, 99394, 118316, 95636, 121084, 66468, 95314, 51880, 68562, 43284, 43446, 27110, 44888, 19814, 20422, 15206, 14496, 7502, 8876, 4222, 5374, 2376, 2390, 808, 2008, 274, 312, 76, 112, 10]
n is: 11, tau is: 2413, and we have: [3475090, 755920, 1278590, 1036826, 1658064, 1041598, 1933438, 1143020, 1880176, 1322744, 1709002, 1090862, 2112490, 1138532, 1462450, 1234316, 1556162, 989470, 1426682, 890202, 1244686, 861630, 962280, 676572, 1037130, 595502, 688432, 542620, 600072, 394014, 529312, 314892, 413768, 262686, 283214, 182652, 260852, 136382, 154338, 105060, 120560, 68362, 85002, 44736, 54548, 34494, 32444, 17900, 31096, 10350, 11190, 6468, 6804, 2534, 5160, 828, 1588, 364, 540, 64, 40]
It took 10.9555 seconds.


Runs all $$\\tau\$$s for $$\n\in {4,5,\dots,11}\$$ in under $$\11\$$ seconds per $$\\tau\$$.
Maybe able to increase that but need to rewrite how the permutations of $$\\sigma\$$ are generated for $$\n=12\$$. The above caches all the permutations but that blows up on my laptop for $$\n=12\$$ due to memory overload.