A128565 -id:A128565 - cp's OEIS frontend

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

Paul D. Hanna, Mar 12 2007

nonn

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

Diagonal of A008302 (Mahonian numbers).
Column 2 of A128564.
Cf. A128565 (column 1), A214086, A048651.

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

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

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

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

Edited by Alois P. Heinz, Mar 05 2013

A128564 Triangle, read by rows, where T(n,k) equals the number of permutations of {1..n+1} with [(nk+k)/2] inversions for n>=k>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 9, 22, 15, 1, 1, 29, 90, 90, 29, 1, 1, 49, 359, 573, 359, 98, 1, 1, 174, 1415, 3450, 3450, 1415, 174, 1, 1, 285, 5545, 17957, 29228, 21450, 5545, 628, 1, 1, 1068, 21670, 110010, 230131, 230131, 110010, 21670, 1068, 1

Offset: 0

Paul D. Hanna, Mar 12 2007

Row sums equal 2*n! for n>0.

			Row sums equal 2*n! for n>0:
[1, 2, 4, 12, 48, 240, 1440, 10080, 80640, ..., 2*n!,...].
Triangle begins:
  1;
  1,    1;
  1,    2,     1;
  1,    5,     5,      1;
  1,    9,    22,     15,       1;
  1,   29,    90,     90,      29,       1;
  1,   49,   359,    573,     359,      98,       1;
  1,  174,  1415,   3450,    3450,    1415,     174,      1;
  1,  285,  5545,  17957,   29228,   21450,    5545,    628,     1;
  1, 1068, 21670, 110010,  230131,  230131,  110010,  21670,  1068,    1;
  1, 1717, 84591, 526724, 1729808, 2409581, 1729808, 686763, 84591, 4015, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A008302 (Mahonian numbers); A128565 (column 1), A128566 (column 2).

Row sums give A098558.

b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
       add(b(u+j-1, o-j)*x^(u+j-1), j=1..o)+
       add(b(u-j, o+j-1)*x^(u-j), j=1..u)))
    end:
T:= (n, k)-> coeff(b(n+1, 0), x, iquo((n+1)*k, 2)):
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, May 02 2017

b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1, Sum[b[u + j - 1, o - j]* x^(u+j-1), {j, 1, o}] + Sum[b[u-j, o+j-1]*x^(u-j), {j, 1, u}]]];
T[n_, k_] := Coefficient[b[n+1, 0], x, Quotient[(n+1)*k, 2]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

{T(n,k)=local(faq=prod(j=1, n+1, (1-q^j)/(1-q))); polcoeff(faq, (n*k+k)\2, q)}

T(n,k) = A008302(n+1, [(nk+k)/2]) = coefficient of q^[(nk+k)/2] in the q-factorial of n+1 for n>=0.

cp's OEIS Frontend

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

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