A200580 Sum of dimension exponents of supercharacter of unipotent upper triangular matrices.

0, 1, 10, 73, 490, 3246, 21814, 150535, 1072786, 7915081, 60512348, 479371384, 3932969516, 33392961185, 293143783762, 2658128519225, 24872012040510, 239916007100054, 2383444110867378, 24363881751014383, 256034413642582418, 2763708806499744097

Offset: 1

Author

Nantel Bergeron, Nov 19 2011

Keywords

• nonn

Comments

Supercharacter theory of unipotent upper triangular matrices over a finite field F(2) is indexed by set partitions S(n) of {1,2,..., n} where a set partition P of {1,2,..., n} is a subset { (i,j) : 1 <= i < j <= n}

such that (i,j) in P implies (i,k),(k,j) are not in P for all i

The dimension of the representation associated to the supercharacter indexed by P is given by 2^Dim(P) where Dim(P) = sum [ j-i , (i,j) in P ].

The sequence we have is a(n) = sum [ Dim(P) , P in S(n) ].

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• M. Aguiar, C. Andre, C. Benedetti, N. Bergeron, Z. Chen, P. Diaconis, A. Hendrickson, S. Hsiao, I.M. Isaacs, A. Jedwab, K. Johnson, G. Karaali, A. Lauve, T. Le, S. Lewis, H. Li, K. Magaard, E. Marberg, J-C. Novelli, A. Pang, F. Saliola, L. Tevlin, J-Y. Thibon, N. Thiem, V. Venkateswaran, C.R. Vinroot, N. Yan and M. Zabrocki, Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras, arXiv:1009.4134 [math.CO], 2010-2011.
• C. André, Basic characters of the unitriangular group, Journal of Algebra, 175 (1995), 287-319.
• B. Chern, P. Diaconis, D. M. Kane and R. C. Rhoades, Closed expressions for averages of set partition statistics, 2013.
• Mikhail Khovanov, Victor Ostrik and Yakov Kononov, Two-dimensional topological theories, rational functions and their tensor envelopes, arXiv:2011.14758 [math.QA], 2020.

Crossrefs

Cf. A011971 (sequence is computed from the Aitken's array b(n,k)

a(n) = sum [ k*(n-k)*b(n,k), k=1..n-1 ]).

Cf. A200660, A200673 (other statistics related to supercharacter theory).

Cf. A000110, A226507.

Programs

Magma

[-2*Bell(n+3)+(n+5)*Bell(n+2): n in [1..30]]; // Vincenzo Librandi, Jul 16 2013

Maple

b:=proc(n,k) option remember;
  if n=1 and k=1 then RETURN(1) fi;
  if k=1 then RETURN(b(n-1,n-1)) fi;
  b(n,k-1)+b(n-1,k-1)
end:
a:=proc(n) local res,k;
  res:=0;
  for k to n-1 do res:=res+k*(n-k)*b(n,k) od;
  res
end:
seq(a(n),n=1..34);

Mathematica

Table[-2 BellB[n+3] + (n+5) BellB[n+2], {n, 1, 30}] (* Vincenzo Librandi, Jul 16 2013 *)

Formula

a(n) = -2*B(n+2) + (n+4)*B(n+1) where B(i) = Bell numbers A000110. [Chern et al.] - N. J. A. Sloane, Jun 10 2013 [for offset 2]

a(n) ~ n^3 * Bell(n) / LambertW(n)^2 * (1 - 2/LambertW(n)). - Vaclav Kotesovec, Jul 28 2021

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