A361952 Array read by antidiagonals: T(n,k) is the number of unlabeled posets with n elements together with a function rk mapping each element to a rank between 1 and k such that whenever v covers w in the poset then rk(v) = rk(w) + 1.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 8, 1, 0, 1, 5, 13, 21, 17, 1, 0, 1, 6, 19, 40, 58, 38, 1, 0, 1, 7, 26, 66, 126, 172, 94, 1, 0, 1, 8, 34, 100, 228, 420, 569, 258, 1, 0, 1, 9, 43, 143, 373, 816, 1537, 2148, 815, 1, 0, 1, 10, 53, 196, 571, 1412, 3140, 6342, 9538, 3038, 1, 0

Offset: 0

Author

Andrew Howroyd, Mar 31 2023

Keywords

• nonn
• tabl

Comments

A poset is counted once for each admissible ranking function. This is an intermediate step in the computation of A361953 where each graded poset is counted exactly once.

Examples

			Array begins:
============================================
n/k| 0 1   2    3    4     5     6     7 ...
---+----------------------------------------
0  | 1 1   1    1    1     1     1     1 ...
1  | 0 1   2    3    4     5     6     7 ...
2  | 0 1   4    8   13    19    26    34 ...
3  | 0 1   8   21   40    66   100   143 ...
4  | 0 1  17   58  126   228   373   571 ...
5  | 0 1  38  172  420   816  1412  2272 ...
6  | 0 1  94  569 1537  3140  5631  9351 ...
7  | 0 1 258 2148 6342 13383 24410 41097 ...
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..860 (first 41 antidiagonals).

Crossrefs

Columns k=0..2 are A000007, A000012, A049312.

Rows n=0..4 are A000012, A000027, A034856, A137742.

The labeled version is A361950.

Cf. A361953.

Programs

PARI

\\ See Links in A361953 for program.
{ my(A=A361952tabl(7)); for(i=1, #A, print(A[i,])) }