If \$R\$ runners were to run a race, in how many orders could they finish such that exactly \$T\$ runners tie?


Given a positive integer \$R\$ and a non-negative integer \$0\leq T\leq {R}\$ produce the number of possible finishing orders of a race with \$R\$ runners of which \$T\$ tied.

Note, however, that runners that tie do not necessarily all tie with each other.

You may accept the number of runners that did not tie, \$R-T\$, in place of either \$R\$ or \$T\$ if you would prefer, just say so in your answer. You may also accept just \$R\$ and output a list of results for \$0\leq T \leq R\$.

This is , so try to make the shortest code in bytes in your language of choice.


1. \$f(R=5, T=0)=120\$

No runners tied and the five runners could have finished in any order, thus \$f(R=5, T=0)=R!=5!=120\$

2. \$f(R=5, T=1)=0\$

There are zero ways for exactly one runner to have tied since ties involve at least two runners.

3. \$f(R=4, T=2)=36\$

  • The first two tied - ** * * - \$\binom{4}{2}\binom{2}{1}\binom{1}{1}=6\times 2\times 1=12\$ ways:
    • AB C D AB D C
      AC B D AC D B
      AD B C AD C B
      BC A D BC D A
      BD A C BD C A
      CD A B CD B A
  • The middle two tied - * ** * - \$\binom{4}{1}\binom{3}{2}\binom{1}{1}=4\times 3\times 1=12\$ ways:
    • A BC D A BD C A CD B
      B AC D B AD C B CD A
      C AB D C AD B C BD A
      D AB C D AC B D BC A
  • The last two tied - * * ** - \$\binom{4}{1}\binom{3}{1}\binom{2}{2}=4\times 3\times 1=12\$ ways:
    • A B CD A C BD A D BC
      B A CD B C AD B D AC
      C A BD C B AD C D AB
      D A BC D B AC D C AB

4. \$f(R=5, T=5)=21\$

  • All five runners tied - ***** - \$\binom{5}{5}=1\$ way
  • The first two and the last three tied - ** *** - \$\binom{5}{2}\binom{3}{3}=10\times 1=10\$ ways:
  • The first three and the last two tied - *** ** - \$\binom{5}{3}\binom{2}{2}=10\times1=10\$ ways:

Test cases

R,T => f(R,T)
1,0 => 1
1,1 => 0
2,0 => 2
2,1 => 0
2,2 => 1
3,0 => 6
3,1 => 0
3,2 => 6
3,3 => 1
4,0 => 24
4,1 => 0
4,2 => 36
4,3 => 8
4,4 => 7
5,0 => 120
5,1 => 0
5,2 => 240
5,3 => 60
5,4 => 100
5,5 => 21
7,5 => 5166

As a table, with x if the input does not need to be handled (all of them would be zero):

T R: 1     2     3     4     5     6     7
0    1     2     6    24   120   720  5040
1    0     0     0     0     0     0     0
2    x     1     6    36   240  1800 15120
3    x     x     1     8    60   480  4200
4    x     x     x     7   100  1170 13440
5    x     x     x     x    21   372  5166
6    x     x     x     x     x   141  3584
7    x     x     x     x     x     x   743

Isomorphic problem

This is the same as \$a(n=R,k=R-T)\$ which is given in A187784 at the Online Encyclopedia of Integer Sequences as the number of ordered set partitions of \$\{1,2,\dots,n\}\$ with exactly \$k\$ singletons.

  • \$\begingroup\$ For cases where T=1, can we output the same as T=0? \$\endgroup\$
    – emanresu A
    Mar 19 at 3:34
  • \$\begingroup\$ @emanresuA no, T=1 is in bounds and should produce zero. \$\endgroup\$ Mar 19 at 4:03
  • \$\begingroup\$ Can I take only \$R\$ and output a list of the answers for all \$0 \leq T \leq R\$? \$\endgroup\$ Mar 19 at 16:55
  • \$\begingroup\$ @CommandMaster I'll allow that (though hopefully not just to avoid a fetch but rather to make for an interesting golf). \$\endgroup\$ Mar 19 at 21:33

8 Answers 8


MATL, 26 24 bytes


Inputs T, then R. The code takes a while for R≥5.

Try it online! You can also see the examples: 1, 2, 3, 4. Or verify test cases for R≤4.


The program consists of three major steps:

  1. Generate all tuples of length R containing (possibly repeated) numbers from the set [1, 2, ..., R].

  2. Keep only those tuples such the tuple contains all numbers [1, 2, ..., u], where u is the number of distinct elements in the tuple. So [1 2 1] would be kept, whereas [1 3 1] or [2 3 2] would be rejected.

  3. For each tuple that survived the previous step, count how many of its R entries are equal to some other entry. Let this count be e. The output of the program is the amount of tuples for which e equals T.


JavaScript (ES6), 88 bytes

Expects (t)(r).


Try it online!


Starting with \$q=r-t\$ and \$p=1\$, we recursively look for all ordered integer partitions of \$r\$. Whenever a new term \$v\$ is added to a given partition \$P\$ of \$r\$:

  • we multiply \$p\$ by the number of ways to choose \$v\$ racers among the remaining ones: $$p\gets p\times\binom{r}{v}$$
  • we subtract \$v\$ from \$r\$
  • we decrement \$q\$ if \$v=1\$

Once a valid partition \$P\$ is complete (ending with \$r=0\$), we test whether we also have \$q=0\$, meaning that there are \$r-t\$ singletons in the finishing order of the race, i.e. \$t\$ ties. If so, we add the product \$p\$ to the final result. Otherwise, we ignore this partition.


PARI/GP, 46 bytes


Attempt This Online!


R, 105 bytes

(or 91 bytes in in R≥4.1 by exchanging function for \)


Attempt This Online!

Input is r=runners, n=non-tied finishers.

Port of Luis Mendo's answer: upvote that one!

R + partitions, 95 bytes

(or 81 bytes in R≥4.1 by exchanging function for \)


Try it at rdrr.io

Input is r=runners, n=non-tied finishers.

Lazily uses partitions library to gather all set partitions, selects those with n singleton sets, and sums the factorials of the number of sets in each partition to account for the number of possible orders.


05AB1E, 21 15 bytes


-6 bytes by porting @LuisMendo's MATL answer, so make sure to upvote him as well!

Inputs in the order \$R,T\$.

Try it online or verify all test cases.

Original 21 bytes answer:


Inputs in the order \$R,T\$.

Brute-force, so pretty slow.

Try it online or verify all test cases.


L         # Push a list in the range [1, first (implicit) input-integer R]
 Zã       # Get the cartesian power of R, creating all possible lists of length R using
          # items of the [1,R]-ranged list, including duplicates
ʒ         # Filter this list of lists by:
 Ð        #  Triplicate the current list
  ê       #  Uniquify and sort the top list
   ā      #  Push a list in the range [1,length] (without popping the list)
    Q     #  Check if the top two lists are equal
 i        #  If this is truthy:
  ¢       #   Pop the remaining two lists, and get the count of each integer
   ≠      #   Check for each count whether it's NOT equal to 1
    O     #   Sum the amount of non-1 counts together
     Q    #   Check if this is equal to the second (implicit) input-integer T
          #  (implicit else: use the current list we've triplicated - which is falsey)
]         # Close both the if-statement and filter
 g        # Pop and push its length to get the amount of remaining valid lists
          # (which is output implicitly as result)
L         # Push a list in the range [1, first (implicit) input-integer R]
 œ        # Get all permutations of this list
  €       # Map over each permutation:
   .œ     #  Get all partitions of this permutation
     €`   # Flatten it one level down to a list of partitions
€         # Map over each partition:
 €        #  Inner map over each part in this partition:
  {       #   Sort the integers in the current part
   Ù      # After the nested map, uniquify the list of partitions with sorted parts
ʒ         # Filter the list of partitions by:
 €g       #  Get the length of each part
   1Ý     #  Push pair [0,1]
     K    #  Remove all 0s and 1s from the list of lengths
      O   #  Sum the remaining lengths (of >=2) together
       Q  #  Check whether this sum is equal to the second (implicit) input-integer `T`
}g        # After the filter: pop and push the length of the remaining partitions
          # (which is output implicitly as result)

Charcoal, 55 bytes


Try it online! Link is to verbose version of code. Takes \$ R \$ and \$ T \$ as parameters. Explanation: Using dynamic programming with a recurrence relation turned out to be a byte shorter than porting @DominicvanEssen's R + partitions answer.


Start with \$ f(0, 0) = 1 \$ and \$ f(0, 1) = 0 \$.


Loop \$ R \$ times.


Calculate \$ f(R, T) \$ using the recurrence relation below and append an extra zero for \$ f(R, R + 1) \$.

$$ f(R, T) = R f(R - 1, T) + \sum _ { i = 2 } ^ T \binom R i f(R - i, T - i) $$


Output \$ f(R, T) \$.


Nekomata + -n, 13 bytes


Attempt This Online!

Takes input as \$R,R-T\$.

Brute force. Very slow.

R                   Range from 1 to the first input (R)
 ↕                  Non-deterministically choose a permutation
  J                 Split the permutation into a list of lists
   :ᵐo=             Check if all the sublists are sorted
       ᵐ#←¬∑        Count the number of sublists with length 1
            =       Check if the number is equal to the second input (R-T)

-n counts the number of solutions.


Wolfram Language (Mathematica), 57 bytes


Try it online!

Accroding to the question description, this is the same as \$a(n=R,k=R-T)\$ which is given in A187784 at the Online Encyclopedia of Integer Sequences as the number of ordered set partitions of \$\{1,2,\dots,n\}\$ with exactly \$k\$ singletons.


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