A361956 Triangle read by rows: T(n,k) is the number of labeled tiered posets with n elements and height k.

1, 0, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 50, 36, 24, 0, 1, 510, 510, 240, 120, 0, 1, 7682, 10620, 4800, 1800, 720, 0, 1, 161406, 312606, 136920, 47040, 15120, 5040, 0, 1, 4747010, 13439076, 5630184, 1678320, 493920, 141120, 40320, 0, 1, 194342910, 821218110, 319384800, 83963880, 21137760, 5594400, 1451520, 362880

Offset: 0

Author

Andrew Howroyd, Apr 02 2023

Keywords

• nonn
• tabl

Comments

A tiered poset is a partially ordered set in which every maximal chain has the same length.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1,      2;
  0, 1,      6,      6;
  0, 1,     50,     36,     24;
  0, 1,    510,    510,    240,   120;
  0, 1,   7682,  10620,   4800,  1800,   720;
  0, 1, 161406, 312606, 136920, 47040, 15120, 5040;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50).

Crossrefs

Row sums are A223911.

Column k=2 is A052332.

Main diagonal is A000142.

The unlabeled version is A361957.

Cf. A222864, A361951.

Programs

PARI

S(M)={my(N=matrix(#M-1, #M-1, i, j, sum(k=1, i-j+1, (2^j-1)^k*M[i-j+1, k])/j!)); for(i=1, #N, for(j=1, i, N[i,j] -= sum(k=1, j-1, N[i-k, j-k]/k!))); N}
C(n)={my(M=matrix(n+1,n+1), R=M); M[1,1]=R[1,1]=1; for(h=1, n, M=S(M); for(i=h, n, R[i+1,h+1] = i!*vecsum(M[i-h+1,]))); R}
{ my(A=C(7)); for(i=1, #A, print(A[i, 1..i])) }