Nantel Bergeron - cp's OEIS frontend

User: Nantel Bergeron

Nantel Bergeron has authored 3 sequences.

A200580 Sum of dimension exponents of supercharacter of unipotent upper triangular matrices.

0, 1, 10, 73, 490, 3246, 21814, 150535, 1072786, 7915081, 60512348, 479371384, 3932969516, 33392961185, 293143783762, 2658128519225, 24872012040510, 239916007100054, 2383444110867378, 24363881751014383, 256034413642582418, 2763708806499744097

Offset: 1

Nantel Bergeron, Nov 19 2011

nonn

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

The dimension of the representation associated to the supercharacter indexed by P is given by 2^Dim(P) where Dim(P) = sum [ j-i , (i,j) in P ].

The sequence we have is a(n) = sum [ Dim(P) , P in S(n) ].

Vincenzo Librandi, Table of n, a(n) for n = 1..200
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 and 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.
B. Chern, P. Diaconis, D. M. Kane and R. C. Rhoades, Closed expressions for averages of set partition statistics, 2013.
Mikhail Khovanov, Victor Ostrik and Yakov Kononov, Two-dimensional topological theories, rational functions and their tensor envelopes, arXiv:2011.14758 [math.QA], 2020.

Cf. A011971 (sequence is computed from the Aitken's array b(n,k)
  a(n) = sum [ k*(n-k)*b(n,k), k=1..n-1 ]).
Cf. A200660, A200673 (other statistics related to supercharacter theory).
Cf. A000110, A226507.

[-2*Bell(n+3)+(n+5)*Bell(n+2): n in [1..30]]; // Vincenzo Librandi, Jul 16 2013

b:=proc(n,k) option remember;
  if n=1 and k=1 then RETURN(1) fi;
  if k=1 then RETURN(b(n-1,n-1)) fi;
  b(n,k-1)+b(n-1,k-1)
end:
a:=proc(n) local res,k;
  res:=0;
  for k to n-1 do res:=res+k*(n-k)*b(n,k) od;
  res
end:
seq(a(n),n=1..34);

Table[-2 BellB[n+3] + (n+5) BellB[n+2], {n, 1, 30}] (* Vincenzo Librandi, Jul 16 2013 *)

a(n) = -2*B(n+2) + (n+4)*B(n+1) where B(i) = Bell numbers A000110. [Chern et al.] - N. J. A. Sloane, Jun 10 2013 [for offset 2]

a(n) ~ n^3 * Bell(n) / LambertW(n)^2 * (1 - 2/LambertW(n)). - Vaclav Kotesovec, Jul 28 2021

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

Original entry on oeis.org

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

Nantel Bergeron, Nov 20 2011

nonn

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).

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.

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

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);

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 *)

A200660 Sum of the number of arcs describing the set partitions of {1,2,...,n}.

Original entry on oeis.org

0, 1, 8, 49, 284, 1658, 9974, 62375, 406832, 2769493, 19668054, 145559632, 1121153604, 8974604065, 74553168520, 641808575961, 5718014325296, 52653303354906, 500515404889978, 4905937052293759, 49530189989912312, 514541524981377909, 5494885265473192914

Offset: 1

Nantel Bergeron, Nov 20 2011

nonn

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 statistics used to compute the supercharacter table is the number of arcs in P (that is, the cardinality |P| of P).
The sequence we have is arcs(n) = Sum_{P in S(n)} |P|.

Seiichi Manyama, Table of n, a(n) for n = 1..500
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, and 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.

Cf. A011971 (sequence is computed from Aitken's array b(n,k) arcs(n) = Sum_{k=1..n-1} k*b(n,k)).
Cf. A200580, A200673 (other statistics related to supercharacter table).
Cf. A367955.

b:=proc(n,k) option remember;
  if n=1 and k=1 then RETURN(1) fi;
  if k=1 then RETURN(b(n-1,n-1)) fi;
  b(n,k-1)+b(n-1,k-1)
end:
arcs:=proc(n) local res,k;
  res:=0;
  for k to n-1 do res:=res+ k*b(n,k) od;
  res
end:
seq(arcs(n),n=1..34);

b[n_, k_] := b[n, k] = Which[n == 1 && k == 1, 1, k == 1, b[n - 1, n - 1], True, b[n, k - 1] + b[n - 1, k - 1]];
arcs[n_] := Module[{res = 0, k}, For[k = 1, k <= n-1, k++, res = res + k * b[n, k]]; res];
Array[arcs, 34] (* Jean-François Alcover, Nov 25 2017, translated from Maple *)

a(n) = Sum_{k=1..n} Stirling2(n,k) * k * (n-k). - Ilya Gutkovskiy, Apr 06 2021

a(n) = Sum_{k=n..n*(n+1)/2} (k-n) * A367955(n,k). - Alois P. Heinz, Dec 11 2023

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User: Nantel Bergeron

A200580 Sum of dimension exponents of supercharacter of unipotent upper triangular matrices.

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

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A200660 Sum of the number of arcs describing the set partitions of {1,2,...,n}.

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Maple

Mathematica

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