cp's OEIS Frontend

A318974 Total number of valid hook configurations of permutations of [n].

%I A318974 #17 Sep 14 2018 12:32:23
%S A318974 1,1,2,6,22,99,520,3126,21164,159226,1318000,11902268,116444668,
%T A318974 1226878267,13849980752,166773534726,2133706472788,28904542964982,
%U A318974 413312731986464,6221110588709700,98321023508946088,1627902016045648206,28178332753660846304
%N A318974 Total number of valid hook configurations of permutations of [n].
%C A318974 a(n) = -k_{n+1}(-1), where k_m(lambda) is the cumulant of the free Poisson law with rate lambda. This is usually defined for lambda > 0, but there is a natural extension to lambda <= 0.
%C A318974 a(n) is the number of pairs (rho,r), where rho is a set partition of {0,...,n} and r is an acyclic orientation of the crossing graph of rho in which the block containing 0 is the only source (see the Josuat-Verges paper or the Defant-Engen-Miller paper for definitions).
%H A318974 Colin Defant, Michael Engen, and Jordan A. Miller, <a href="https://arxiv.org/abs/1809.01340">Stack-sorting, set partitions, and Lassalle's sequence</a>, arXiv:1809.01340 [math.CO], 2018.
%H A318974 Matthieu Josuat-Verges, <a href="https://arxiv.org/abs/1203.3157">Cumulants of the q-semicircular law, Tutte polynomials, and heaps</a>, arXiv:1203.3157 [math.CO], 2012.
%F A318974 Let m_n(lambda) = Sum_{k=1..n} lambda^k * A001263(n,k). If we define k_n(lambda) by Sum_{n>=1} k_n(lambda) * z^n/n! = log(1 + Sum_{n>=1} m_n(lambda) * z^n/n!), then a(n) = -k_{n+1}(-1).
%F A318974 Define E(m,n) by E(n,n) = 1 and E(m,n) = Sum_{j=1..m} Sum_{i=1.. n-m-1} binomial(n-m-1,i-1) * F_j(i+j-1) * F_{m-j}(n-j-i) for 0 <= m < n, where F_m(n) = Sum_{j=m..n} E_j(n). Then a(n) = F_0(n).
%t A318974 Table[(-(m + 1)!) SeriesCoefficient[Log[1 + Sum[Sum[(1/n) Binomial[n, k] Binomial[n, k - 1] (-1)^k (z^n/n!), {k, 1, n}], {n, 1, 100}]], {z, 0, m + 1}], {m, 1, 10}]
%Y A318974 Cf. A001263, A180874.
%K A318974 easy,nonn
%O A318974 1,3
%A A318974 _Colin Defant_, Sep 06 2018