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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)



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) ].


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, 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, R. C. Rhoades, Closed expressions for averages of set partition statistics, 2013.


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


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)


a:=proc(n) local res, k;


  for k to n-1 do res:=res+k*(n-k)*b(n, k) od;



seq(a(n), n=1..34);


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


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


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




Nantel Bergeron, Nov 19 2011



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Last modified March 20 15:18 EDT 2018. Contains 300985 sequences. (Running on oeis4.)