A316983 Number of non-isomorphic self-dual multiset partitions of weight n.

1, 1, 2, 4, 9, 17, 36, 72, 155, 319, 677, 1429, 3094, 6648, 14518, 31796, 70491, 156818, 352371, 795952, 1813580, 4155367, 9594425, 22283566, 52122379, 122631874, 290432439, 691831161, 1658270316, 3997272089, 9692519896, 23631827354, 57943821449, 142834652193

Offset

0,3

Comments

Also the number of nonnegative integer square symmetric matrices with sum of elements equal to n, under row and column permutations.

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Example

Non-isomorphic representatives of the a(4) = 9 self-dual multiset partitions:
  (1111),
  (1)(222), (2)(122), (11)(22), (12)(12),
  (1)(1)(23), (1)(2)(33), (1)(3)(23),
  (1)(2)(3)(4).
The a(4) = 9 square symmetric matrices:
. [4]
.
. [3 0]  [2 0]  [2 1]  [1 1]
. [0 1]  [0 2]  [1 0]  [1 1]
.
. [2 0 0]  [1 1 0]  [0 1 1]
. [0 1 0]  [1 0 0]  [1 0 0]
. [0 0 1]  [0 0 1]  [1 0 0]
.
. [1 0 0 0]
. [0 1 0 0]
. [0 0 1 0]
. [0 0 0 1]

Prog

(PARI) vector(25, n, n--; T(n, n)) \\ T(n, k) defined in A318805. - Andrew Howroyd, Jan 16 2024

Crossrefs

Row sums of A320796.

Main diagonal of A318805.

Cf. A000009, A001055, A007716, A007717, A020555, A045778.

Cf. A316974, A316978, A316979, A316980, A316981.

Sequence in context: A245122 A291728 A268649 * A136326 A362033 A059973

Adjacent sequences: A316980 A316981 A316982 * A316984 A316985 A316986

Keyword

nonn

Author

Gus Wiseman, Jul 18 2018

Extensions

Terms a(9) and beyond from Andrew Howroyd, Sep 03 2018

Status

approved