login
A200673
Total number of nested arcs in the set partitions of n.
2
0, 0, 0, 1, 16, 170, 1549, 13253, 110970, 928822, 7862353, 67758488, 596837558, 5385257886, 49837119320, 473321736911, 4614233950422, 46168813528478, 474017189673555, 4992024759165631, 53902161267878974
OFFSET
1,5
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 statistic used to compute the supercharacter table is the number of nested pair in P. That is the cardinality nst(P)= | { (i < r < s < j : (i,j),(r,s) in P } |.
The sequence we have is nst(n) = sum [ nst(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, 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.
MAPLE
c:=proc(n, k, j) option remember;
if n=3 and k=2 and j=1 then RETURN(1) fi;
if k=2 and j=1 then RETURN(c(n-1, n-2, 1)) fi;
if k=j+1 then RETURN(c(n, j+1, j-1) + c(n-1, j, j-1)) fi;
c(n, k-1, j)+c(n-1, k-1, j)
end:
nst:=proc(n) local res, k, j;
res:=0;
for j to n-3 do
for k from j+1 to n-2 do
res:=res+j*(k-j)*c(n, k, j) od; od;
res
end:
seq(nst(n), n=1..21);
MATHEMATICA
c[n_, k_, j_] := c[n, k, j] = Which[n == 3 && k == 2 && j == 1, 1, k == 2 && j == 1, c[n - 1, n - 2, 1], k == j + 1, c[n, j + 1, j - 1] + c[n - 1, j, j - 1], True, c[n, k - 1, j] + c[n - 1, k - 1, j]];
nst[n_] := Module[{res = 0, k, j}, For[j = 1, j <= n - 3, j++, For[k = j + 1, k <= n - 2, k++, res = res + j*(k - j)*c[n, k, j]]]; res];
Array[nst, 21] (* Jean-François Alcover, Nov 25 2017, translated from Maple *)
CROSSREFS
Cf. A200580, A200660 (other statistics related to supercharacter table).
Sequence in context: A048557 A174645 A021424 * A230510 A238725 A221789
KEYWORD
nonn
AUTHOR
Nantel Bergeron, Nov 20 2011
STATUS
approved