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.
One of the statistics used to compute the supercharacter table is the number of arcs in P (that is, the cardinality |P| of P).
The sequence we have is arcs(n) = Sum_{P in S(n)} |P|.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..500
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.
FORMULA
a(n) = Sum_{k=1..n} Stirling2(n,k) * k * (n-k). - Ilya Gutkovskiy, Apr 06 2021
a(n) = Sum_{k=n..n*(n+1)/2} (k-n) * A367955(n,k). - Alois P. Heinz, Dec 11 2023
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:
arcs:=proc(n) local res, k;
res:=0;
for k to n-1 do res:=res+ k*b(n, k) od;
res
end:
seq(arcs(n), n=1..34);
MATHEMATICA
b[n_, k_] := b[n, k] = Which[n == 1 && k == 1, 1, k == 1, b[n - 1, n - 1], True, b[n, k - 1] + b[n - 1, k - 1]];
arcs[n_] := Module[{res = 0, k}, For[k = 1, k <= n-1, k++, res = res + k * b[n, k]]; res];
Array[arcs, 34] (* Jean-François Alcover, Nov 25 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Nantel Bergeron, Nov 20 2011
STATUS
approved
