A104779 a(n) is the sum of entries of n-th Kostka matrix for the partitions of n.

1, 1, 3, 7, 21, 57, 182, 565, 1931, 6670, 24537, 92337, 364602, 1477148, 6219031, 26875932, 119930947, 548688443, 2580814003, 12425175838, 61302331782, 309055818656, 1592723862598, 8374123173858, 44917765035082, 245452258746785, 1366116578058731, 7736098938006873

Offset: 0

Author

Alford Arnold, Mar 24 2005

Keywords

• nonn
• changed

Comments

a(n) is the number of symmetric nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and weakly decreasing row and column sums. - Ludovic Schwob, Aug 29 2023

Examples

			For n=4, {1,1,1,1,1} + {0,1,1,2,3} + {0,0,1,1,2} + {0,0,0,1,3} + {0,0,0,0,1} = 21.
		

Links

• Ludovic Schwob, Table of n, a(n) for n = 0..39
• E. Egge et al., From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix
• E. Egge et al., From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix, European Journal of Combinatorics, Volume 31, Issue 8, December 2010, Pages 2014-2027.
• Wouter Meeussen, Schur Polynomials
• Wouter Meeussen, Kostka numbers up to partitions of 20
• Wouter Meeussen, Mathematica code for 'kostka' function
• Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 13.

Crossrefs

Cf. A000041, A000085, A005651, A036038, A097522, A104707, A178718.

Row sums of A104778.

Programs

Mathematica

(* See Meeussen link. *)

Formula

Row sums of A104778.

Extensions

a(7) corrected by Alford Arnold, Dec 31 2010

a(8)-a(21) from Amiram Eldar, May 03 2024