A381427 -id:A381427 - cp's OEIS frontend

A381426 A(n,k) is the sum over all ordered partitions of [n] of k^j for an ordered partition with j inversions; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 13, 8, 1, 1, 5, 36, 75, 16, 1, 1, 6, 79, 696, 541, 32, 1, 1, 7, 148, 3851, 27808, 4683, 64, 1, 1, 8, 249, 14808, 567733, 2257888, 47293, 128, 1, 1, 9, 388, 44643, 5942608, 251790113, 369572160, 545835, 256, 1, 1, 10, 571, 113480, 40065301, 9546508128, 335313799327, 121459776768, 7087261, 512

Offset: 0

Alois P. Heinz, Feb 23 2025

			Square array A(n,k) begins:
   1,    1,       1,         1,          1,            1,             1, ...
   1,    1,       1,         1,          1,            1,             1, ...
   2,    3,       4,         5,          6,            7,             8, ...
   4,   13,      36,        79,        148,          249,           388, ...
   8,   75,     696,      3851,      14808,        44643,        113480, ...
  16,  541,   27808,    567733,    5942608,     40065301,     199246816, ...
  32, 4683, 2257888, 251790113, 9546508128, 179833594207, 2099255895008, ...

Alois P. Heinz, Antidiagonals n = 0..55, flattened
Wikipedia, Inversion
Wikipedia, Partition of a set

Columns k=0-9 give: A011782, A000670, A289545, A347841, A347842, A347843, A385408, A347844, A347845, A347846.
Main diagonal gives A381427.
Cf. A381299, A381369.

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

b[o_, u_, t_, k_] := b[o, u, t, k] = If[u + o == 0, 1, If[t == 1, b[u + o, 0, 0, k], 0] + Sum[If[k == u + j - 1 == 0, 1, k^(u + j - 1)]*b[o - j, u + j - 1, 1, k], {j, 1, o}]];
A[n_, k_] := b[n, 0, 0, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Apr 19 2025, after Alois P. Heinz *)

A(n,k) = Sum_{j=0..binomial(n,2)} k^j * A381299(n,j).

A381373 Sum over all partitions of [n] of n^j for a partition with j inversions.

Original entry on oeis.org

1, 1, 2, 7, 72, 3276, 915848, 2011878835, 42723411900032, 10608257527069388539, 35808039364308986083608352, 1828963737334508176477805993389490, 1618534282345584818909121118371843799592960, 28472613161534902071627567919297331348486838233018341

Offset: 0

Alois P. Heinz, Feb 21 2025

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..41
Wikipedia, Inversion
Wikipedia, Partition of a set

Main diagonal of A381369.

Cf. A062173, A120325, A125810, A381427.

b:= proc(o, u, t, k) option remember;
     `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2, k), 0)+add(k^(u+j-1)*
        b(o-j, u+j-1, min(2, t+1), k), j=`if`(t=0, 1, 1..o)))
    end:
a:= n-> b(n, 0$2, n):
seq(a(n), n=0..15);

b[o_, u_, t_, k_] := b[o, u, t, k] =
   If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0, k], 0] + Sum[k^(u + j - 1)*
   b[o - j, u + j - 1, Min[2, t + 1], k], {j, If[t == 0, {1}, Range[o]]}]];
a[n_] := b[n, 0, 0, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 15 2025, after Alois P. Heinz *)

a(n) = Sum_{j>=0} n^j * A125810(n,j).
a(n) = A381369(n,n).
a(n) mod n = A062173(n) for n>=1.
a(n) mod 2 = A120325(n+1) for n>=1.

cp's OEIS Frontend

A381426 A(n,k) is the sum over all ordered partitions of [n] of k^j for an ordered partition with j inversions; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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