A085642 Number of columns in the character table of the symmetric group S_n that have zero sum.

0, 1, 1, 2, 3, 6, 8, 12, 17, 26, 35, 49, 66, 92, 121, 161, 211, 280, 360, 466, 596, 766, 968, 1225, 1538, 1935, 2408, 2996, 3707, 4588, 5636, 6918, 8456, 10329, 12552, 15236, 18431, 22275, 26817, 32242, 38661, 46306, 55294, 65942, 78464, 93252, 110561

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 11 2003

Keywords

• nonn

Comments

Conjecture: Equals the number of partitions of n with at least one part congruent to 2 mod 4. - Vladeta Jovovic, Jul 12 2003. This conjecture was established by Christine Bessenrodt and Jorn B. Olsson (olsson(AT)math.ku.dk), Sep 13 2004.

Also number of partitions of n with some odd part repeated. - Vladeta Jovovic, Feb 05 2005

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
• 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. 6.
• Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, On the sum of the entries in a character table, Proc. 36th Conf. Formal Power Series Alg. Comb., Sem. Lotharingien Comb (2024) Vol. 91B, Art. No. 99.
• Christine Bessenrodt and Jorn Olsson, On the sequence A085642
• Dominique Gouyou-Beauchamps and Philippe Nadeau, Signed Enumeration of Ribbon Tableaux with Local Rules and Generalizations of the Schensted Correspondence, in Formal Power Series and Algebraic Combinatorics, Nankai University, Tianjin, China, 2007.
• Dominique Gouyou-Beauchamps and Philippe Nadeau, Signed enumeration of ribbon tableaux: an approach through growth diagrams, Journal of Algebraic Combinatorics, 2011; DOI 10.1007/s10801-011-0324-2.

Crossrefs

Cf. A006950, A082733.

Programs

Mathematica

Rest[PartitionsP[Range[0,47]] - CoefficientList[Series[Product[(1+x^(2 k - 1))/(1 - x^(2 k)), {k,48}], {x,0,47}], x]] (* Wouter Meeussen, Dec 20 2017 *)

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jul 11 2018

Extensions

Corrected and extended by Vladeta Jovovic, Jul 12 2003

More terms from David Wasserman, Feb 08 2005