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

1,3

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<l<j.

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.

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

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 *)

Prog

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

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.

Sequence in context: A016211 A055424 A243878 * A181678 A206817 A159687

Adjacent sequences: A200577 A200578 A200579 * A200581 A200582 A200583

Keyword

nonn

Author

Nantel Bergeron, Nov 19 2011

Status

approved