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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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..21.

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

Adjacent sequences: A200670 A200671 A200672 * A200674 A200675 A200676

KEYWORD

nonn

AUTHOR

Nantel Bergeron, Nov 20 2011

STATUS

approved

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Last modified November 28 02:51 EST 2022. Contains 358406 sequences. (Running on oeis4.)