A001833 Number of labeled graded partially ordered sets with n elements. (Formerly M3067 N1243)

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

Offset

0,3

Comments

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.

References

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).

Links

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

Example

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.

Prog

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

Crossrefs

Row sums of A361951.

Graded posets with no chain of length 3 are counted by A001831.

Cf. A223911, A228551, A361920 (unlabeled version).

Sequence in context: A135749 A005647 A158876 * A001035 A267634 A277407

Adjacent sequences: A001830 A001831 A001832 * A001834 A001835 A001836

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved