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A200660 Sum of the number of arcs describing the set partitions of {1,2,...,n}. 4
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
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
a(n) = Sum_{k=n..n*(n+1)/2} (k-n) * A367955(n,k). - Alois P. Heinz, Dec 11 2023
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).
Cf. A367955.
Sequence in context: A026774 A089383 A351128 * A028443 A001108 A115598
KEYWORD
nonn
AUTHOR
Nantel Bergeron, Nov 20 2011
STATUS
approved

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Last modified March 29 10:59 EDT 2024. Contains 371277 sequences. (Running on oeis4.)