A334622 A(n,k) is the sum of the k-th powers of the descent set statistics for permutations of [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 8, 1, 1, 2, 10, 24, 16, 1, 1, 2, 18, 88, 120, 32, 1, 1, 2, 34, 360, 1216, 720, 64, 1, 1, 2, 66, 1576, 14460, 24176, 5040, 128, 1, 1, 2, 130, 7224, 190216, 994680, 654424, 40320, 256, 1, 1, 2, 258, 34168, 2675100, 46479536, 109021500, 23136128, 362880, 512

Offset

0,6

Links

Alois P. Heinz, Antidiagonals n = 0..25, flattened

R. Ehrenborg and A. Happ, On the powers of the descent set statistic, arXiv:1709.00778 [math.CO], 2017.

Formula

A(n,k) = Sum_{j=0..ceiling(2^(n-1))-1} A060351(n,j)^k.

Example

Square array A(n,k) begins:
   1,   1,     1,      1,        1,          1,            1, ...
   1,   1,     1,      1,        1,          1,            1, ...
   2,   2,     2,      2,        2,          2,            2, ...
   4,   6,    10,     18,       34,         66,          130, ...
   8,  24,    88,    360,     1576,       7224,        34168, ...
  16, 120,  1216,  14460,   190216,    2675100,     39333016, ...
  32, 720, 24176, 994680, 46479536, 2368873800, 128235838496, ...
  ...

Maple

b:= proc(u, o, t) option remember; expand(`if`(u+o=0, 1,
      add(b(u-j, o+j-1, t+1)*x^floor(2^(t-1)), j=1..u)+
      add(b(u+j-1, o-j, t+1), j=1..o)))
    end:
A:= (n, k)-> (p-> add(coeff(p, x, i)^k, i=0..degree(p)))(b(n, 0$2)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

b[u_, o_, t_] := b[u, o, t] = Expand[If[u + o == 0, 1,
    Sum[b[u - j, o + j - 1, t + 1] x^Floor[2^(t - 1)], {j, 1, u}] +
    Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]]];
A[n_, k_] := Function[p, Sum[Coefficient[p, x, i]^k, {i, 0, Exponent[p, x]}]][b[n, 0, 0]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

Crossrefs

Columns k=0-4 give: A011782, A000142, A060350, A291902, A291903.

Rows n=0+1, 2-3 give: A000012, A007395(k+1), A052548(k+1).

Main diagonal gives A334623.

Cf. A060351, A335545.

Sequence in context: A373778 A110971 A136788 * A136450 A355011 A131054

Adjacent sequences: A334619 A334620 A334621 * A334623 A334624 A334625

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 09 2020

Status

approved