A321884 Number A(n,k) of partitions of n into colored blocks of equal parts with colors from a set of size k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 4, 3, 0, 1, 4, 6, 8, 5, 0, 1, 5, 8, 15, 14, 7, 0, 1, 6, 10, 24, 27, 24, 11, 0, 1, 7, 12, 35, 44, 51, 40, 15, 0, 1, 8, 14, 48, 65, 88, 93, 64, 22, 0, 1, 9, 16, 63, 90, 135, 176, 159, 100, 30, 0, 1, 10, 18, 80, 119, 192, 295, 312, 264, 154, 42, 0
Offset: 0
Examples
A(3,2) = 8: 3a, 3b, 2a1a, 2a1b, 2b1a, 2b1b, 111a, 111b. Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, 6, 7, 8, ... 0, 2, 4, 6, 8, 10, 12, 14, 16, ... 0, 3, 8, 15, 24, 35, 48, 63, 80, ... 0, 5, 14, 27, 44, 65, 90, 119, 152, ... 0, 7, 24, 51, 88, 135, 192, 259, 336, ... 0, 11, 40, 93, 176, 295, 456, 665, 928, ... 0, 15, 64, 159, 312, 535, 840, 1239, 1744, ... 0, 22, 100, 264, 544, 970, 1572, 2380, 3424, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..200, flattened
- Jessica Jay and Benjamin Lees, Combinatorial identities from an inhomogeneous Ising chain, arXiv:2401.16311 [math.PR], 2024.
- Wikipedia, Partition (number theory)
Crossrefs
Programs
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Maple
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add( (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k))) end: A:= (n, k)-> b(n$2, k): seq(seq(A(n, d-n), n=0..d), d=0..14); -
Mathematica
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]][n - i j], {j, 1, n/i}] k + b[n, i - 1, k]]]; A[n_, k_] := b[n, n, k]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)
Formula
G.f. of column k: Product_{j>=1} (1+(k-1)*x^j)/(1-x^j).
A(n,k) = Sum_{i=0..floor((sqrt(1+8*k)-1)/2)} k!/(k-i)! * A321878(n,i).