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A222866 Triangle T(n,k) of weakly graded (3+1)-free partially ordered sets (posets) on n labeled vertices with height k. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,tabl
AUTHOR
Joel B. Lewis, Mar 07 2013
STATUS
approved

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Last modified February 23 18:28 EST 2024. Contains 370283 sequences. (Running on oeis4.)