A369919 Triangular array read by rows. T(n,k) is the number of labeled posets on [n] of rank at most one with exactly k elements of positive indegree, n >= 0, 0 <= k <= max{0,n-1}.

1, 1, 1, 2, 1, 9, 3, 1, 28, 54, 4, 1, 75, 490, 270, 5, 1, 186, 3375, 6860, 1215, 6, 1, 441, 20181, 118125, 84035, 5103, 7, 1, 1016, 111132, 1668296, 3543750, 941192, 20412, 8, 1, 2295, 580644, 21003948, 116363646, 95681250, 9882516, 78732, 9

Offset

0,4

Comments

The rank of a poset is the number of cover relations in a maximal chain.

Equivalently, T(n,k) is the number of labeled posets P on [n] of rank at most one such that |image(P)| = k.

Links

Table of n, a(n) for n=0..45.

Formula

E.g.f.: Sum_{n>=0} x^n/n!*exp(y*x)^(2^n-1).

T(n,1) = A058877(n).

Example

Triangle begins
  1;
  1;
  1,   2;
  1,   9,    3;
  1,  28,   54,    4;
  1,  75,  490,  270,    5;
  1, 186, 3375, 6860, 1215, 6;
  ...

Mathematica

nn = 9; Map[Select[#, # > 0 &] &, Table[n!, {n, 0, nn}] CoefficientList[Series[ Sum[ Exp[y  x]^(2^n - 1)  x^n/n!, {n, 0, nn}], {x, 0, nn}], {x, y}]] // Grid

Crossrefs

Cf. A001831 (row sums), A058877, A263859, A369921.

Sequence in context: A223141 A021460 A090884 * A355881 A095888 A160510

Adjacent sequences: A369916 A369917 A369918 * A369920 A369921 A369922

Keyword

nonn,tabl

Author

Geoffrey Critzer, Feb 05 2024

Status

approved