A361953 Triangle read by rows: T(n,k) is the number of unlabeled weakly graded (ranked) posets with n elements and rank k.

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 8, 6, 1, 0, 1, 20, 30, 9, 1, 0, 1, 55, 145, 66, 12, 1, 0, 1, 163, 745, 465, 111, 15, 1, 0, 1, 556, 4245, 3444, 964, 165, 18, 1, 0, 1, 2222, 27880, 28024, 8618, 1652, 228, 21, 1, 0, 1, 10765, 218058, 259974, 83322, 16569, 2556, 300, 24, 1

Offset

0,9

Comments

Here weakly graded means that there exists a rank function rk from the poset to the integers such that whenever v covers w in the poset, we have rk(v) = rk(w) + 1.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..860 (rows 0..40)

Andrew Howroyd, PARI Program, Apr 2023.

Wikipedia, Graded poset.

Formula

G.f. of column k >= 2: C(k,x)/C(k-1,x) - C(k-1,x)/C(k-2,x) where C(k,x) is the g.f. of column k of A361952.

Example

Triangle begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   3,    1;
  0, 1,   8,    6,    1;
  0, 1,  20,   30,    9,   1;
  0, 1,  55,  145,   66,  12,   1;
  0, 1, 163,  745,  465, 111,  15,  1;
  0, 1, 556, 4245, 3444, 964, 165, 18, 1;
  ...

Prog

(PARI) \\ See link for program code.
{ my(A=A361953tabl(8)); for(i=1, #A, print(A[i, 1..i])) }

Crossrefs

Row sums are A361920.

The labeled version is A361951.

Cf. A263859, A361952, A361954 (connected).

Sequence in context: A274494 A274490 A193357 * A112413 A294219 A091480

Adjacent sequences: A361950 A361951 A361952 * A361954 A361955 A361956

Keyword

nonn,tabl

Author

Andrew Howroyd, Mar 31 2023

Status

approved