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A222864 Triangle T(n,k) of strongly graded (3+1)-free partially ordered sets (posets) on n labeled vertices with height k. 2
1, 1, 2, 1, 6, 6, 1, 50, 36, 24, 1, 510, 510, 240, 120, 1, 7682, 7380, 4800, 1800, 720, 1, 161406, 141246, 91560, 47040, 15120, 5040, 1, 4747010, 3444756, 2162664, 1134000, 493920, 141120, 40320, 1, 194342910, 110729310, 61286400, 32253480, 14605920, 5594400 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Here "strongly graded" means that every maximal chain has the same length. Alternate terminology includes "graded" (e.g., in Stanley 2011) and "tiered" (as in A006860). 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.

EXAMPLE

For n = 3, there is 1 strongly graded poset of height 1 (the antichain), 6 strongly graded posets of height 2, and 6 strongly graded posets of height 3 (the chains), and all of these are (3+1)-free. Thus, the third row of the triangle is 1, 6, 6.

CROSSREFS

For row-sums (strongly graded (3+1)-free posets with n labeled vertices, disregarding height), see A222863. For weakly graded (3+1)-free posets, see A222865. For all strongly graded posets, see A006860. For all (3+1)-free posets, see A079145.

Sequence in context: A008297 A090582 A079641 * A232433 A271881 A182729

Adjacent sequences:  A222861 A222862 A222863 * A222865 A222866 A222867

KEYWORD

nonn,tabl

AUTHOR

Joel B. Lewis, Mar 07 2013

STATUS

approved

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Last modified September 25 23:23 EDT 2020. Contains 337346 sequences. (Running on oeis4.)