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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 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.)