A361920 -id:A361920 - cp's OEIS frontend

A001833 Number of labeled graded partially ordered sets with n elements.

1, 1, 3, 19, 219, 3991, 106623, 3964339, 199515459, 13399883551, 1197639892983, 143076298623259, 23053861370437659, 5062745845287855271, 1530139311543346178223, 641441466132460086890179, 375107113287994040621904819, 307244526491924695346004951151, 353511145615118063468292270299943

Offset: 0

N. J. A. Sloane

nonn

Here "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. Note that this notion of grading is weaker than in sequence A006860, which counts posets in which all maximal chains have the same length.

			The poset on {a, b, c, d, e} defined by the relations a < b < c and d < e is counted by this sequence. (For example, one associated rank function is rk(a) = rk(d) = 0, rk(b) = rk(e) = 1 and rk(c) = 2.) However, the poset defined by the relations a < b < c and a < d < e < c is not graded and so not counted by this sequence.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..100
David A. Klarner, The number of graded partially ordered sets, Journal of Combinatorial Theory, vol.6, no.1, pp.12-19, (January-1969).
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
Index entries for sequences related to posets

Row sums of A361951.
Graded posets with no chain of length 3 are counted by A001831.
Cf. A223911, A228551, A361920 (unlabeled version).

\\ C(n) is defined in A361951.
seq(n)={my(c=C(n)); Vec(serlaplace(c[n+1]/c[n]))} \\ Andrew Howroyd, Mar 31 2023

Corrected and edited by Joel B. Lewis, Mar 28 2011
a(7)-a(15) from Daniele P. Morelli, Aug 25 2013
a(16)-a(18) from Sean A. Irvine, Sep 25 2015

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

Original entry on oeis.org

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.

A361912 The number of unlabeled graded posets with n elements.

Original entry on oeis.org

1, 1, 2, 4, 10, 28, 93, 354, 1621, 9110, 64801, 595976, 7204091, 115561423, 2473540433, 70853213144, 2720354016419, 140170631441858, 9702605436760235, 903309202327818566, 113234129823368903523, 19137461395401601912043, 4366007821745938984134203

Offset: 0

Martin Rubey, Mar 29 2023

nonn

A partially ordered set is graded if all maximal chains have the same length. This is called tiered by some authors.

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

Row sums of A361957.

Cf. A000112, A223911 (labeled), A001833, A361920, A361959 (connected).

\\ See PARI link in A361957 for program code.
A361912seq(20) \\ Andrew Howroyd, Apr 03 2023

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

Terms a(8) and beyond from Andrew Howroyd, Mar 30 2023

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.

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A001833 Number of labeled graded partially ordered sets with n elements.

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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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A361912 The number of unlabeled graded posets with n elements.

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

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