A054946 Number of strongly connected labeled tournaments on n nodes.
1, 0, 2, 24, 544, 22320, 1677488, 236522496, 64026088576, 33832910196480, 35262092417856512, 72926863133112198144, 300318571786159783496704, 2467430973323656141183549440, 40490606137578335674252914280448
Offset: 1
Examples
For n=3, there are two minimal idempotent generating sets for the semigroup of singular mappings on {1,2,3}. Writing (a,b,c) to indicate the map for which 1->a, etc, the relevant generating sets are: {(1,1,3),(1,2,2),(3,2,3)} and {(2,2,3),(1,3,3),(1,2,1)}.
References
- Archer, K., Gessel, I. M., Graves, C., & Liang, X. (2020). Counting acyclic and strong digraphs by descents. Discrete Mathematics, 343(11), 112041.
Links
- N. J. A. Sloane, Table of n, a(n) for n = 1..80 [Shortened file because terms grow rapidly: see Sloane link below for additional terms]
- James East and Robert D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359, 2014
- J. M. Howie, Idempotent generators in finite full transformation semigroups, Proc. Roy. Soc. Edinburgh Sect. A, 81 (1978), no. 3-4, 317-323.
- Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See p. 33.
- Valery A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.
- Thierry Monteil and Khaydar Nurligareev, Asymptotic probability for connectedness, arXiv:2401.00818 [math.CO], 2024. See p. 12.
- N. J. A. Sloane, Table of n, a(n) for n = 1..100
- E. M. Wright, The number of irreducible tournaments, Glasgow Math. J., 11 (1970), 97-101.
Programs
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Maple
with(powseries): powcreate(t(n)=2^(n*(n-1)/2)/n!): s := evalpow(1-1/t): a := tpsform(s, x, 21): for n from 0 to 20 do printf(`%d,`,n!*coeff(a,x,n)) od: f:=array(0..500); F:=array(0..500); M:=100; f[1]:=1; F[1]:=1; lprint(1,f[1]); for n from 2 to M do F[n]:=2^(n*(n-1)/2); f[n]:=F[n]-add( binomial(n,s)*f[s]*F[n-s], s=1..n-1); lprint(n,f[n]); od:
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Mathematica
F[n_] := 2^(n*(n - 1)/2); a[1] = 1; a[n_] := a[n] = F[n] - Sum[Binomial[n, s]*a[s]*F[n-s], {s, 1, n-1 }]; Array[a, 15] (* Jean-François Alcover, Nov 13 2017, from first formula *) -
PARI
seq(n)={my(a=vector(n)); for(n=1, n, a[n] = 2^(n*(n-1)/2) - sum(k=1, n-1, binomial(n,k)*2^((n-k)*(n-k-1)/2)*a[k])); a} \\ Andrew Howroyd, Jan 10 2022
Formula
Let F(n) = 2^(n*(n-1)/2). Then a(n) is defined by the recurrence a(1)=1, F(n) = a(n) + Sum_{s=1..n-1} binomial(n,s)*a(s)*F(n-s). [Wright]
E.g.f.: 1-1/(1+f(x)) where f(x) = Sum_{m>=1} 2^(m(m-1)/2) x^m / m!.
Wright also gives an asymptotic expansion for a(n).
Comments