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%I #14 Nov 26 2017 14:01:07
%S 0,0,0,1,16,170,1549,13253,110970,928822,7862353,67758488,596837558,
%T 5385257886,49837119320,473321736911,4614233950422,46168813528478,
%U 474017189673555,4992024759165631,53902161267878974
%N Total number of nested arcs in the set partitions of n.
%C 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.
%C 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 } |.
%C The sequence we have is nst(n) = sum [ nst(P), P in S(n) ].
%H 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, <a href="http://arxiv.org/abs/1009.4134">Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras</a>, arXiv:1009.4134 [math.CO], 2010-2011.
%H C. André, <a href="https://doi.org/10.1006/jabr.2001.8734">Basic characters of the unitriangular group</a>, Journal of Algebra, 175 (1995), 287-319.
%p c:=proc(n,k,j) option remember;
%p if n=3 and k=2 and j=1 then RETURN(1) fi;
%p if k=2 and j=1 then RETURN(c(n-1,n-2,1)) fi;
%p if k=j+1 then RETURN(c(n,j+1,j-1) + c(n-1,j,j-1)) fi;
%p c(n,k-1,j)+c(n-1,k-1,j)
%p end:
%p nst:=proc(n) local res,k,j;
%p res:=0;
%p for j to n-3 do
%p for k from j+1 to n-2 do
%p res:=res+j*(k-j)*c(n,k,j) od; od;
%p res
%p end:
%p seq(nst(n),n=1..21);
%t 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]];
%t 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];
%t Array[nst, 21] (* _Jean-François Alcover_, Nov 25 2017, translated from Maple *)
%Y Cf. A200580, A200660 (other statistics related to supercharacter table).
%K nonn
%O 1,5
%A _Nantel Bergeron_, Nov 20 2011
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