A335545 -id:A335545 - cp's OEIS frontend

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

Alois P. Heinz, Sep 09 2020

			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, ...
  ...

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.

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.

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);

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 *)

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

A168562 Sum of squares of Eulerian numbers in row n of triangle A008292 with a(0)=1.

Original entry on oeis.org

1, 1, 2, 18, 244, 5710, 188908, 8702820, 524888040, 40393084950, 3853034107900, 446671026849916, 61824801560228056, 10072685383683311116, 1907978676359896992824, 415795605119514578204616, 103294156408291202467520976, 29018125910193347265466916070

Offset: 0

Paul D. Hanna, Nov 29 2009

nonn

Row sums of Eulerian triangle A008292 yield the factorials.

			a(1) = 1 = 1;
a(2) = 1 + 1 = 2;
a(3) = 1 + 4^2 + 1 = 18;
a(4) = 1 + 11^2 + 11^2 + 1 = 244;
a(5) = 1 + 26^2 + 66^2 + 26^2 + 1 = 5710;
a(6) = 1 + 57^2 + 302^2 + 302^2 + 57^2 + 1 = 188908.

Alois P. Heinz, Table of n, a(n) for n = 0..253

Cf. A008292.

Column k=2 of A335545.

a:= n-> add(combinat[eulerian1](n, k)^2, k=0..n):
seq(a(n), n=0..18);  # Alois P. Heinz, Sep 10 2020

{a(n)=sum(k=0,n,sum(j=0, k, (-1)^j*(k-j)^n*binomial(n+1, j))^2)}

a(n) = Sum_{k=0..n} [ Sum_{j=0..k} (-1)^j*(k-j)^n*C(n+1, j) ]^2.

A335546 Sum of the n-th powers of the (positive) number of permutations of [n] with j descents (j=0..max(0,n-1)).

Original entry on oeis.org

1, 1, 2, 66, 29284, 1276095330, 1517369276209828, 487282023126556473747120, 7083859535826403131254334882236808, 55967865856667137512789954604119317025544887810, 29848842646338160810499934016489864718841567718650088748120500

Offset: 0

Alois P. Heinz, Sep 12 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..30

Main diagonal of A335545.

Cf. A008292, A173018, A334623, A335545.

B[n_, k_] := Sum[(-1)^j Binomial[n+1, j] (k-j+1)^n, {j, 0, k+1}];
a[n_] := If[n == 0, 1, Sum[B[n, j]^n, {j, 0, n-1}]];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Sep 03 2021, from A335545 *)

a(n) = A335545(n,n).

a(n) = Sum_{j=0..n} A173018(n,j)^n.

cp's OEIS Frontend

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.

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A168562 Sum of squares of Eulerian numbers in row n of triangle A008292 with a(0)=1.

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A335546 Sum of the n-th powers of the (positive) number of permutations of [n] with j descents (j=0..max(0,n-1)).

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