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JavaScript (ES6), 154 bytes

g=(a,n,l)=>l?g(a,a.reduce((r,_,i)=>r+=i<=l&a.indexOf(l,i)<0,n*l--),l):n
f=(n,i=0,a=[],j=i-n%++i)=>n?f(n/i|0,i,[...a.map(e=>e+=e>=j),j]):g(a,0,i-1)

<input type=number min=0 oninput=o.textContent=f(+this.value)><pre id=o>

0-indexed. Works by converting the index to a permutation, inverting it, then converting back to an index. g inverts the permutation as it converts it back to an index. Ungolfed:

function index(n) {
    var a = [0];
    for (var i = 1; n = Math.floor(n / i); i++) {
        var j = i - n % (i + 1);
        for (var k = 0; k < i; k++) {
            if (a[k] > j) a[k]++;
        }
        a.push(j);
    }
    a = [...a.keys()].map(k => a.indexOf(k));
    while (i) {
        n *= i--;
        j = a.pop();
        for (k = 0; k < i; k++) {
            if (a[k] > j) n++;
        }
    }
    return n;
}