A221492 -id:A221492 - cp's OEIS frontend

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

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

nonn

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.

Cf. A079145 (labeled semitransitive orders), A000112.

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 *)

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]

More terms from Mathieu Guay-Paquet, Jan 18 2013

A221493 Number of tangled bicolored graphs on n labeled vertices.

Original entry on oeis.org

0, 0, 0, 0, 12, 120, 2460, 64680, 2323692, 111920760, 7272700860, 639739653960, 76764606923532, 12645557866982040, 2878366780307114460, 909775941009828296040, 401039212596034472197932, 247339947733328456032703160, 214013123181627427780427544060

Offset: 0

Mathieu Guay-Paquet, Jan 18 2013

A bicolored graph on n labeled vertices, k of which are black, and (n-k) of which are white, can be represented as a k X (n-k) matrix, where the (i,j) entry is 1 if the i-th black vertex is adjacent to the j-th white vertex, and 0 otherwise. Then, the graph is tangled if (1) the matrix does not have any rows or columns of all 0's or all 1's; and (2) it is not possible to permute the rows of the matrix and the columns of the matrix to obtain a matrix of the form
[ A | J ]
[---+---]
[ 0 | B ]
where the top right block J consists of all 1's, and the bottom left block 0 consists of all 0's.

			The only tangled bicolored graph on 4 vertices (up to isomorphism) consists of 2 black vertices, 2 white vertices, and 2 edges, with each black vertex joined to a different white vertex. Given 4 labels, there are 12 distinct ways of labeling the vertices, so a(4) = 1.

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

Cf. A221492, A079145.

nmax = 19;
B[x_] = Sum[Exp[2^n x] x^n/n!, {n, 0, nmax}] + O[x]^nmax;
T[x_] = 2 Exp[-x] - 1 - 1/B[x] + O[x]^nmax;
CoefficientList[T[x], x] Range[0, nmax-1]! (* Jean-François Alcover, Aug 12 2018 *)

E.g.f.: T(x) = 2*e^(-x) - 1 - 1/B(x), where B(x) is the e.g.f. for A047863.

A221494 Table read by downward diagonals: T(n,k) = number of skeleta of (3+1)-free posets with n clone sets and k tangles.

Original entry on oeis.org

1, 1, 1, 3, 5, 1, 12, 28, 16, 2, 55, 165, 152, 47, 4, 273, 1001, 1265, 658, 136, 9, 1428, 6188, 9919, 7315, 2547, 392, 21, 7752, 38760, 75208, 71981, 35975, 9252, 1130, 51, 43263, 245157, 558144, 657356, 431599, 159701, 32286, 3262, 127, 246675, 1562275

Offset: 0

Mathieu Guay-Paquet, Jan 18 2013

			There are 28 skeleta of (3+1)-free posets with 1 clone set and 2 tangles.
Table begins
  1   1    3     12      55      273 ...
  1   5   28    165    1001     6188 ...
  1  16  152   1265    9919    75208 ...
  2  47  658   7315   71981   657356 ...
  4 136 2547  35975  431599  4660516 ...
  9 392 9252 159701 2277821 28589750 ...
  ......................................

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

Cf. A079145, A079146, A221492, A221493

G.f.: S(x, y) is the unique power series solution of the equation S(x, y) = 1 + S(x, y)^2 * x / (1 + x) + S(x, y)^3 * y.

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

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A221493 Number of tangled bicolored graphs on n labeled vertices.

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A221494 Table read by downward diagonals: T(n,k) = number of skeleta of (3+1)-free posets with n clone sets and k tangles.

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