login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A200660 Sum of the number of arcs describing the set partitions of {1,2,...,n}. 3
0, 1, 8, 49, 284, 1658, 9974, 62375, 406832, 2769493, 19668054, 145559632, 1121153604, 8974604065, 74553168520, 641808575961, 5718014325296, 52653303354906, 500515404889978, 4905937052293759, 49530189989912312, 514541524981377909, 5494885265473192914 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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 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|.

LINKS

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.

FORMULA

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

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

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).

Sequence in context: A026719 A026774 A089383 * A028443 A001108 A115598

Adjacent sequences:  A200657 A200658 A200659 * A200661 A200662 A200663

KEYWORD

nonn

AUTHOR

Nantel Bergeron, Nov 20 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 24 09:43 EST 2022. Contains 350534 sequences. (Running on oeis4.)