A381369 A(n,k) is the sum over all partitions of [n] of k^j for a partition with j inversions; 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, 5, 8, 1, 1, 2, 6, 15, 16, 1, 1, 2, 7, 28, 52, 32, 1, 1, 2, 8, 47, 204, 203, 64, 1, 1, 2, 9, 72, 628, 2344, 877, 128, 1, 1, 2, 10, 103, 1552, 17327, 43160, 4140, 256, 1, 1, 2, 11, 140, 3276, 84416, 1022983, 1291952, 21147, 512
Offset: 0
Examples
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, 5, 6, 7, 8, 9, 10, ... 8, 15, 28, 47, 72, 103, 140, ... 16, 52, 204, 628, 1552, 3276, 6172, ... 32, 203, 2344, 17327, 84416, 307867, 915848, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..60, flattened
- Wikipedia, Inversion
- Wikipedia, Partition of a set
Crossrefs
Programs
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Maple
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, k)-> b(n, 0$2, k): seq(seq(A(n, d-n), n=0..d), d=0..10); -
Mathematica
Unprotect[Power]; 0^0 = 1; Protect[Power]; 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_, k_] := b[n, 0, 0, k]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Mar 15 2025, after Alois P. Heinz *)
Formula
A(n,k) = Sum_{j>=0} k^j * A125810(n,j).