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%I #29 Dec 11 2018 08:58:46
%S 1,1,3,13,111,1381,22383,461413,12163791,420626821,19880808303,
%T 1337330559973,130909732781391,18649561895661061,3830195104867879023,
%U 1124247654215697637093,469367653568553278229711,278046313987470874905216901,233462156432002170491075384943
%N Strongly graded (3+1)-free partially ordered sets (posets) on n labeled vertices.
%C Here "strongly graded" means that every maximal chain has the same length. Alternate terminology includes "graded" (e.g., in Stanley 2012) and "tiered" (as in A006860). A poset is said to be (3+1)-free if it does not contain four elements a, b, c, d such that a < b < c and d is incomparable to the other three.
%D R. P. Stanley, Enumerative Combinatorics, Volume 1. Cambridge University Press. 2nd edition, 2012. http://math.mit.edu/~rstan/ec/ec1/
%H J. B. Lewis and Y. X. Zhang, <a href="http://arxiv.org/abs/1106.5480">Enumeration of Graded (3+1)-Avoiding Posets</a>, J. Combin. Theory Ser. A 120 (2013), no. 6, 1305-1327.
%F G.f.: H(e^x, Psi(x)) where H(x, y) = 1 + (2x^3 - 3x^2 + (x^3 - 4x^2 + 4x)y)/(2x^2 + x + (x^2 - 2x - 1)y) and Psi(x) is the g.f. for A047863.
%t m = maxExponent = 19;
%t Psi[x_] = Sum[E^(2^n*x)*x^n/n!, {n, 0, m}] + O[x]^m;
%t H[x_, y_] = 1+(2x^3 - 3x^2 + (x^3 - 4x^2 + 4x)y)/(2x^2 + x + (x^2-2x-1) y);
%t CoefficientList[H[E^x, Psi[x]] + O[x]^m, x] Range[0, m-1]! (* _Jean-François Alcover_, Dec 11 2018 *)
%Y For strongly graded (3+1)-free posets by height, see A222864. For weakly graded (3+1)-free posets, see A222865. For all strongly graded posets, see A006860. For all (3+1)-free posets, see A079145.
%K nonn
%O 0,3
%A _Joel B. Lewis_, Mar 07 2013
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