A200673 Total number of nested arcs in the set partitions of n.

0, 0, 0, 1, 16, 170, 1549, 13253, 110970, 928822, 7862353, 67758488, 596837558, 5385257886, 49837119320, 473321736911, 4614233950422, 46168813528478, 474017189673555, 4992024759165631, 53902161267878974, 596448192670732180, 6760141422115666131, 78438566784031690720

Offset: 1

Author

Nantel Bergeron, Nov 20 2011

Keywords

• nonn

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 < k < j.

One of the statistic used to compute the supercharacter table is the number of nested pairs 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_{P in S(n)} nst(P).

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.

Crossrefs

Cf. A200580, A200660 (other statistics related to supercharacter table).

Programs

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 *)