A361955 -id:A361955 - cp's OEIS frontend

A361920 Number of unlabeled ranked posets with n elements.

1, 1, 2, 5, 16, 61, 280, 1501, 9394, 68647, 591570, 6108298, 77162708, 1219779207, 24648006828, 647865966973, 22437052221282, 1032905858402302, 63591727342096158, 5258562027225785955, 586001891321599337103, 88241281449605821921186, 17996565026907866304071630

Offset: 0

Martin Rubey, Mar 29 2023

nonn

A partially ordered set is ranked if there is a function from the poset elements to the integers such that the function value of a covering element is precisely one larger than the function value of the covered element. This is called graded by some authors.

			For n=5, A000112(n) - a(n) = 63 - 61 = 2 because we have 2 posets with 5 elements that are not ranked: a<b<c<d  a<e<d  and  a<c<e  a<d  b<d  b<e where < means "is covered by". - _Geoffrey Critzer_, Oct 29 2023

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

Row sums of A361953.

Cf. A000112, A001833 (labeled), A223911, A361912, A361955.

\\ See PARI link in A361953 for program code.
A361920seq(20) \\ Andrew Howroyd, Apr 01 2023

sum(1 for P in posets(n) if P.is_ranked())

Terms a(8) and beyond from Andrew Howroyd, Mar 31 2023

A361954 Triangle read by rows: T(n,k) is the number of unlabeled connected weakly graded (ranked) posets with n elements and rank k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 10, 23, 8, 1, 0, 27, 107, 56, 11, 1, 0, 88, 557, 388, 98, 14, 1, 0, 328, 3271, 2888, 839, 149, 17, 1, 0, 1460, 22424, 23900, 7512, 1470, 209, 20, 1, 0, 7799, 183273, 226807, 73405, 14715, 2308, 278, 23, 1

Offset: 1

Andrew Howroyd, Mar 31 2023

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.

			Triangle begins:
  1;
  0,   1;
  0,   2,    1;
  0,   4,    5,    1;
  0,  10,   23,    8,   1;
  0,  27,  107,   56,  11,   1;
  0,  88,  557,  388,  98,  14,  1;
  0, 328, 3271, 2888, 839, 149, 17, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..820 (rows 1..40)
Wikipedia, Graded poset.

Column k=2 is A007776.
Row sums are A361955.
Cf. A342500, A361953 (not necessarily connected).

\\ See PARI link in A361953 for program code.
{ my(A=A361954tabl(8)); for(i=1, #A, print(A[i, 1..i])) }

cp's OEIS Frontend

A361920 Number of unlabeled ranked posets with n elements.

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