|
|
A200580
|
|
Sum of dimension exponents of supercharacter of unipotent upper triangular matrices.
|
|
4
|
|
|
0, 1, 10, 73, 490, 3246, 21814, 150535, 1072786, 7915081, 60512348, 479371384, 3932969516, 33392961185, 293143783762, 2658128519225, 24872012040510, 239916007100054, 2383444110867378, 24363881751014383, 256034413642582418, 2763708806499744097
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
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.
|
|
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).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|