A321880 -id:A321880 - cp's OEIS frontend

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

Alois P. Heinz, Aug 27 2019

			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, ...

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)

Columns k=0-4 give: A000007, A000041, A015128, A264686, A266821.
Main diagonal gives A321880.
Cf, A000142, A003056, A321878.

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);

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 *)

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).

A325916 Number of partitions of n into colored blocks of equal parts with colors from a set of size n such that the block with largest parts has the first color.

Original entry on oeis.org

1, 1, 2, 5, 11, 27, 76, 177, 428, 966, 2724, 5986, 14322, 31241, 68632, 174364, 374901, 841417, 1792950, 3803764, 7688426, 18376432, 37158444, 80078021, 163155272, 335521478, 658661436, 1298215354, 2820956914, 5523327097, 11240000648, 22117134452, 43666070406

Offset: 0

Alois P. Heinz, Sep 08 2019

nonn

			a(3) = 5: 3a, 2a1a, 2a1b, 2a1c, 111a.

Alois P. Heinz, Table of n, a(n) for n = 0..1650

Cf. A321880.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, k*add(
      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i) +b(n, i-1, k)))
    end:
a:= n-> `if`(n=0, 1, b(n$3)/n):
seq(a(n), n=0..34);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, k Sum[With[{t = n - i j},  b[t, Min[t, i - 1], k]], {j, 1, n/i}] + b[n, i - 1, k]]];
a[n_] := If[n == 0, 1, b[n, n, n]/n];
a /@ Range[0, 34] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)

a(n) = 1/n * [x^n] Product_{j=1..n} (1+(n-1)*x^j)/(1-x^j) for n>0, a(0)=1.

a(n) = A321880(n)/n for n > 0, a(0) = 1.

cp's OEIS Frontend

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.

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