A244581 Multisets of multisets corresponding to integer partitions lambda, drawn from |lambda| symbols, where the sizes of the multisets are given by the elements of lambda as is the total number of occurrences of each symbol.

1, 1, 1, 1, 2, 1, 1, 4, 2, 2, 1, 1, 7, 6, 4, 3, 2, 1, 1, 11, 17, 5, 8, 12, 2, 4, 3, 2, 1, 1, 16, 41, 23, 15, 39, 14, 8, 8, 12, 4, 4, 3, 2, 1, 1, 22, 87, 86, 17, 26, 108, 81, 27, 18, 16, 40, 15, 17, 3, 8, 12, 4, 4, 3, 2, 1, 1, 29, 167, 263, 109, 42, 263, 342, 78, 81, 115, 10, 31, 116, 87, 60, 39, 11, 16, 40, 15, 17, 5, 8, 12, 4, 4, 3, 2, 1, 1, 37, 296, 695, 509, 73, 64, 578, 1177, 602, 216, 525, 169, 64, 57, 306, 380, 90, 189, 261, 34, 38, 26, 32, 117, 88, 61, 40, 22, 3, 16, 40, 15, 17, 5, 8, 12, 4, 4, 3, 2, 1

Offset

1,5

Comments

The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. The row lengths of this irregular table are given by the partition function A000041.

References

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Links

Marko Riedel, Table of n, a(n) for n = 1..271

Marko Riedel et. al, Partitioning a multiset into multisets of fixed sizes

Marko Riedel, Maple code for sequence using cycle index of the symmetric group and cycle index substitution.

Wikipedia, Young's lattice

Formula

With the partition given by Prod_{k=1}^l A_k^{tau_k} the closed form is [Prod_{k=1}^l A_k^{tau_k}] Prod_{k=1}^l Z(S_{tau_k}; Z(S_k; Prod_{k'=1}^l A_k')) where Z(S_k) is the cycle index of the symmetric group.

Example

With the partition [1,1,2] or A_1 A_2 A_3^2 we get four multisets: {{A_1}, {A_2}, {A_3^2}}, {{A_1}, {A_3}, {A_2 A_3}}, {{A_2}, {A_3}, {A_1, A_3}} and {{A_3}, {A_3}, {A_1 A_2}}.
The initial list of the partitions is:
  1;
  1,1; 2;
  1,1,1; 1,2; 3;
  1,1,1,1; 1,1,2; 2+2; 1+3; 4;
The data then yields the following values:
  1,
  1, 1,
  1, 2, 1,
  1, 4, 2, 2.

Crossrefs

Sequence in context: A029268 A347629 A176452 * A064191 A127420 A129033

Adjacent sequences: A244578 A244579 A244580 * A244582 A244583 A244584

Keyword

tabf,nonn

Author

Marko Riedel, Jul 31 2018

Status

approved