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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.

%I A334622 #25 Dec 20 2020 12:35:53
%S A334622 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,
%T A334622 2,34,360,1216,720,64,1,1,2,66,1576,14460,24176,5040,128,1,1,2,130,
%U A334622 7224,190216,994680,654424,40320,256,1,1,2,258,34168,2675100,46479536,109021500,23136128,362880,512
%N 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.
%H A334622 Alois P. Heinz, <a href="/A334622/b334622.txt">Antidiagonals n = 0..25, flattened</a>
%H A334622 R. Ehrenborg and A. Happ, <a href="https://arxiv.org/abs/1709.00778">On the powers of the descent set statistic</a>, arXiv:1709.00778 [math.CO], 2017.
%F A334622 A(n,k) = Sum_{j=0..ceiling(2^(n-1))-1} A060351(n,j)^k.
%e A334622 Square array A(n,k) begins:
%e A334622    1,   1,     1,      1,        1,          1,            1, ...
%e A334622    1,   1,     1,      1,        1,          1,            1, ...
%e A334622    2,   2,     2,      2,        2,          2,            2, ...
%e A334622    4,   6,    10,     18,       34,         66,          130, ...
%e A334622    8,  24,    88,    360,     1576,       7224,        34168, ...
%e A334622   16, 120,  1216,  14460,   190216,    2675100,     39333016, ...
%e A334622   32, 720, 24176, 994680, 46479536, 2368873800, 128235838496, ...
%e A334622   ...
%p A334622 b:= proc(u, o, t) option remember; expand(`if`(u+o=0, 1,
%p A334622       add(b(u-j, o+j-1, t+1)*x^floor(2^(t-1)), j=1..u)+
%p A334622       add(b(u+j-1, o-j, t+1), j=1..o)))
%p A334622     end:
%p A334622 A:= (n, k)-> (p-> add(coeff(p, x, i)^k, i=0..degree(p)))(b(n, 0$2)):
%p A334622 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A334622 b[u_, o_, t_] := b[u, o, t] = Expand[If[u + o == 0, 1,
%t A334622     Sum[b[u - j, o + j - 1, t + 1] x^Floor[2^(t - 1)], {j, 1, u}] +
%t A334622     Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]]];
%t A334622 A[n_, k_] := Function[p, Sum[Coefficient[p, x, i]^k, {i, 0, Exponent[p, x]}]][b[n, 0, 0]];
%t A334622 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Dec 20 2020, after _Alois P. Heinz_ *)
%Y A334622 Columns k=0-4 give: A011782, A000142, A060350, A291902, A291903.
%Y A334622 Rows n=0+1, 2-3 give: A000012, A007395(k+1), A052548(k+1).
%Y A334622 Main diagonal gives A334623.
%Y A334622 Cf. A060351, A335545.
%K A334622 nonn,tabl
%O A334622 0,6
%A A334622 _Alois P. Heinz_, Sep 09 2020