A222866 Triangle T(n,k) of weakly graded (3+1)-free partially ordered sets (posets) on n labeled vertices with height k.

1, 1, 2, 1, 12, 6, 1, 86, 84, 24, 1, 840, 1110, 480, 120, 1, 11642, 16620, 9120, 3240, 720, 1, 227892, 300846, 185640, 82320, 25200, 5040, 1, 6285806, 6810804, 4299624, 2142000, 816480, 221760, 40320, 1, 243593040, 199239270, 117205200, 60890760, 26157600

Offset

1,3

Comments

Here "weakly graded" means that there is a rank function rk from the vertices to the integers such that whenever x covers y we have rk(x) = rk(y) + 1. Alternate terminology includes "graded" and "ranked." A poset is said to be (3+1)-free if it does not contain four elements a, b, c, d such that a < b < c and d is incomparable to the other three.

Links

Joel B. Lewis, Rows n = 1..20 of triangle, flattened

J. B. Lewis and Y. X. Zhang, Enumeration of Graded (3+1)-Avoiding Posets, J. Combin. Theory Ser. A 120 (2013), no. 6, 1305-1327.

Formula

G.F. is given in the Lewis-Zhang paper.

Crossrefs

For row-sums (weakly graded (3+1)-free posets with n labeled vertices, disregarding height), see A222865. For strongly graded (3+1)-free posets, see A222863. For all weakly graded posets, see A001833. For all (3+1)-free posets, see A079145.

Sequence in context: A048743 A049055 A276998 * A342587 A008285 A119274

Adjacent sequences: A222863 A222864 A222865 * A222867 A222868 A222869

Keyword

nonn,tabl

Author

Joel B. Lewis, Mar 07 2013

Status

approved