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

Author

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

Keywords

• nonn

Links

• 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.

Crossrefs

Cf. A079145 (labeled semitransitive orders), A000112.

Programs

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 *)

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]

Extensions

More terms from Mathieu Guay-Paquet, Jan 18 2013