A361954 -id:A361954 - cp's OEIS frontend

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

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, 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;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..860 (rows 0..40)
Andrew Howroyd, PARI Program, Apr 2023.
Wikipedia, Graded poset.

Row sums are A361920.
The labeled version is A361951.
Cf. A263859, A361952, A361954 (connected).

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

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.

A361955 Number of unlabeled connected weakly graded (ranked) posets with n elements.

Original entry on oeis.org

1, 1, 1, 3, 10, 42, 202, 1146, 7493, 56996, 508609, 5414635, 70214227, 1134439731, 23331152887, 621768153861, 21761221300058, 1009759125475973, 62534859409597022, 5193886959561972984, 580677490292990902682, 87649885799470898359728, 17907726747155924589913398

Offset: 0

Andrew Howroyd, Mar 31 2023

nonn

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.

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

Row sums of A361954.

Cf. A000608, A361920, A361953.

\\ See PARI link in A361953 for program code.
A361955seq(20)

Inverse Euler transform of A361920.

A361958 Triangle read by rows: T(n,k) is the number of connected unlabeled tiered posets with n elements and height k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 3, 1, 0, 10, 10, 4, 1, 0, 27, 34, 16, 5, 1, 0, 88, 146, 66, 23, 6, 1, 0, 328, 739, 326, 106, 31, 7, 1, 0, 1460, 4671, 1932, 566, 156, 40, 8, 1, 0, 7799, 37088, 14104, 3577, 878, 217, 50, 9, 1, 0, 51196, 376524, 128652, 27563, 5740, 1276, 290, 61, 10, 1

Offset: 1

Andrew Howroyd, Apr 03 2023

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

			Triangle begins:
  1;
  0,    1;
  0,    2,    1;
  0,    4,    3,    1;
  0,   10,   10,    4,   1;
  0,   27,   34,   16,   5,   1;
  0,   88,  146,   66,  23,   6,  1;
  0,  328,  739,  326, 106,  31,  7, 1;
  0, 1460, 4671, 1932, 566, 156, 40, 8, 1;
  ...

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

Row sums are A361959.

Cf. A361954, A361957 (not necessarily connected).

\\ See PARI link in A361957 for program code.
{ my(A=A361958tabl(9)); for(i=1, #A, print(A[i, 1..i])) }

Column k is the inverse Euler transform of column k of A361957.

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

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A361955 Number of unlabeled connected weakly graded (ranked) posets with n elements.

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A361958 Triangle read by rows: T(n,k) is the number of connected unlabeled tiered posets with n elements and height k.

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