A128566 Number of permutations of {1..n} with n inversions.

1, 0, 0, 1, 5, 22, 90, 359, 1415, 5545, 21670, 84591, 330121, 1288587, 5032235, 19664205, 76893687, 300895513, 1178290263, 4617369760, 18106447251, 71048746505, 278966179936, 1095987764828, 4308300939450, 16944940572831, 66680029591816, 262519664110588

Offset: 0

Author

Paul D. Hanna, Mar 12 2007

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1665

Crossrefs

Diagonal of A008302 (Mahonian numbers).

Column 2 of A128564.

Cf. A128565 (column 1), A214086, A048651.

Programs

Maple

a:= n-> coeff(series(mul((1-q^j)/(1-q), j=1..n), q, n+1), q, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 05 2013

Mathematica

Table[SeriesCoefficient[QPochhammer[x, x, n]/(1-x)^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, May 13 2016 *)

PARI

{a(n)=polcoeff(prod(j=1, n, (1-q^j)/(1-q)),n,q)}

Formula

a(n) = A008302(n,n) = coefficient of q^n in the q-factorial of n.

a(n) = T(n,n) with T(n,k) = T(n-1,k) + Sum_{j=1..n-1} T(n-1,k-j) for n>=0, k>0; T(n,k) = 0 for n<0; T(n,0) = 1 for n>=0. - Alois P. Heinz, Mar 07 2013

a(n) ~ c * 2^(2*n-1) / sqrt(Pi*n), where c = A048651 = QPochhammer[1/2] = 0.28878809508660242127889972192923... . - Vaclav Kotesovec, Sep 07 2014

Extensions

Edited by Alois P. Heinz, Mar 05 2013