A082733 Sum of all entries in character table of the symmetric group S_n.

1, 2, 5, 13, 31, 89, 259, 842, 2810, 10020, 37266, 145373, 586071, 2453927, 10590180, 47159351, 215706629, 1013916313, 4882544468, 24087770591, 121481296510, 626169893024, 3293432146879, 17670096206819, 96589760733604, 537731396393480, 3045955783377644

Offset: 1

Author

Vladeta Jovovic, May 20 2003

Keywords

• nonn

Examples

			a(3) = 5 because the character table of S_3 is / 1 1 1 / 2 0 -1 / 1 -1 1 /.
		

Links

• Ludovic Schwob, Table of n, a(n) for n = 1..200
• Ron M. Adin, Alexander Postnikov, and Yuval Roichman! Combinatorial Gelfand models, J. Algebra, 320(3):1311-1325, 2008.
• Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv preprint arXiv:2406.06036 [math.RT], 2024. See p. 5.
• Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, On the sum of the entries in a character table, Proceedings of the 36th Conference on Formal Power Series and Algebraic Combinatorics (Bochum), Séminaire Lotharingien de Combinatoire 91B (2024), Article #99, 12 pp.
• Joseph Ben Geloun and Sanjaye Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv preprint arXiv:1307.6490 [hep-th], 2013.
• Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 5.

Crossrefs

Cf. A086808, A085624, A085646.

Programs

GAP

A082733 := n -> Sum(Sum(Irr(CharacterTable("Symmetric", n)))); # Eric M. Schmidt, Jul 03 2012

Mathematica

a[n_] := FiniteGroupData[{"SymmetricGroup", n}, "CharacterTable"] // Flatten // Total;
Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 10}] (* Jean-François Alcover, Nov 03 2018 *)

Formula

Let D(x) = Sum_{n>=0} (2n-1)!!*x^n = 1/(1 - x/(1 - 2x/(1 - 3x/...))) and R_r(x) = Sum_{n>=0} o_r(n)*x^n = 1/(1 - x - r*x^2/(1 - x - 2*r*x^2/(1 - x - 3*r*x^2/...))), where o_r(n) = Sum_{k=0..n/2} binomial(n, 2k)*(2k-1)!!*r^k. Then the generating function of this sequence is Sum_{n>=0} a(n)*x^n = Product_{i >= 1} (D(2ix^{4i}) * R_{2i-1}(x^{2i-1}). - Arvind Ayyer, Jun 11 2024

Extensions

More terms from Eric M. Schmidt, Jul 03 2012