A079146 Number of unlabeled semitransitive orders on n elements: (1+3)-free posets.

1, 2, 5, 15, 49, 173, 639, 2469, 9997, 43109, 205092, 1153646, 8523086, 91156133, 1446766659, 32998508358, 1047766596136, 45632564217917, 2711308588849394, 219364550983697100, 24151476334929009951, 3618445112608409433287

Offset

1,2

Links

Table of n, a(n) for n=1..22.

M. Guay-Paquet, A modular relation for the chromatic symmetric functions of (3+1)-free posets, arXiv preprint arXiv:1306.2400 [math.CO], 2013.

M. Guay-Paquet, A. H. Morales, E. Rowland, Structure and enumeration of (3+1)-free posets (extended abstract), arXiv:1212.5356 [math.CO], 2012.

Formula

G.f.: S(x/(1-x), T(x)), where S(x, y) is the g.f. for A221494 and T(x) is the g.f. for A221492. [Mathieu Guay-Paquet, Jan 18 2013]

Mathematica

nmax = 23; co = Coefficient; ex = Exponent;
b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Flatten[Table[Function[ {p}, p + j x^i] /@ b[n - i j, i - 1], {j, 0, n/i}]]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j] co[s, x, i] co[t, x, j], {j, 1, ex[t, x]}], {i, 1, ex[s, x]}]/Product[i^co[s, x, i]*co[s, x, i]!, {i, 1, ex[s, x]}]/Product[i^co[t, x, i] co[t, x, i]!, {i, 1, ex[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
A[d_] := Sum[A[n, d - n], {n, 0, d}];
B[x_] = Sum[A[n] x^n, {n, 0, nmax}];
S[_, _] = 0; Do[S[c_, t_] = Series[1 + (c/(1 + c)) S[c, t]^2 + t S[c, t]^3, {c, 0, nmax}, {t, 0, nmax}] // Normal, {nmax}];
T[x_] = 1 - S[x/(1 - x), 1 - 2x - 1/B[x]];
Rest[CoefficientList[-T[x] + O[x]^nmax, x]] (* Jean-François Alcover, Aug 11 2018, after Alois P. Heinz *)

Crossrefs

Cf. A079145 (labeled semitransitive orders), A000112.

Sequence in context: A149941 A149942 A149943 * A000734 A148366 A005751

Adjacent sequences: A079143 A079144 A079145 * A079147 A079148 A079149

Keyword

nonn

Author

Detlef Pauly (dettodet(AT)yahoo.de), Dec 27 2002

Extensions

More terms from Mathieu Guay-Paquet, Jan 18 2013

Status

approved