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 ]).
KEYWORD
nonn
AUTHOR
Nantel Bergeron, Nov 19 2011
STATUS
approved