cp's OEIS Frontend

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

%I A200580 #39 Sep 08 2022 08:46:00
%S A200580 0,1,10,73,490,3246,21814,150535,1072786,7915081,60512348,479371384,
%T A200580 3932969516,33392961185,293143783762,2658128519225,24872012040510,
%U A200580 239916007100054,2383444110867378,24363881751014383,256034413642582418,2763708806499744097
%N A200580 Sum of dimension exponents of supercharacter of unipotent upper triangular matrices.
%C A200580 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}
%C A200580   such that (i,j) in P implies (i,k),(k,j) are not in P for all i<l<j.
%C A200580 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 ].
%C A200580 The sequence we have is a(n) = sum [ Dim(P) , P in S(n) ].
%H A200580 Vincenzo Librandi, <a href="/A200580/b200580.txt">Table of n, a(n) for n = 1..200</a>
%H A200580 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, <a href="http://arxiv.org/abs/1009.4134">Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras</a>, arXiv:1009.4134 [math.CO], 2010-2011.
%H A200580 C. André, <a href="https://doi.org/10.1006/jabr.2001.8734">Basic characters of the unitriangular group</a>, Journal of Algebra, 175 (1995), 287-319.
%H A200580 B. Chern, P. Diaconis, D. M. Kane and R. C. Rhoades, <a href="http://math.stanford.edu/~rhoades/FILES/setpartitions.pdf">Closed expressions for averages of set partition statistics</a>, 2013.
%H A200580 Mikhail Khovanov, Victor Ostrik and Yakov Kononov, <a href="https://arxiv.org/abs/2011.14758">Two-dimensional topological theories, rational functions and their tensor envelopes</a>, arXiv:2011.14758 [math.QA], 2020.
%F A200580 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]
%F A200580 a(n) ~ n^3 * Bell(n) / LambertW(n)^2 * (1 - 2/LambertW(n)). - _Vaclav Kotesovec_, Jul 28 2021
%p A200580 b:=proc(n,k) option remember;
%p A200580   if n=1 and k=1 then RETURN(1) fi;
%p A200580   if k=1 then RETURN(b(n-1,n-1)) fi;
%p A200580   b(n,k-1)+b(n-1,k-1)
%p A200580 end:
%p A200580 a:=proc(n) local res,k;
%p A200580   res:=0;
%p A200580   for k to n-1 do res:=res+k*(n-k)*b(n,k) od;
%p A200580   res
%p A200580 end:
%p A200580 seq(a(n),n=1..34);
%t A200580 Table[-2 BellB[n+3] + (n+5) BellB[n+2], {n, 1, 30}] (* _Vincenzo Librandi_, Jul 16 2013 *)
%o A200580 (Magma) [-2*Bell(n+3)+(n+5)*Bell(n+2): n in [1..30]]; // _Vincenzo Librandi_, Jul 16 2013
%Y A200580 Cf. A011971 (sequence is computed from the Aitken's array b(n,k)
%Y A200580   a(n) = sum [ k*(n-k)*b(n,k), k=1..n-1 ]).
%Y A200580 Cf. A200660, A200673 (other statistics related to supercharacter theory).
%Y A200580 Cf. A000110, A226507.
%K A200580 nonn
%O A200580 1,3
%A A200580 _Nantel Bergeron_, Nov 19 2011