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

%I #23 Jun 24 2017 01:02:22

%S 1,1,2,1,6,6,1,50,36,24,1,510,510,240,120,1,7682,7380,4800,1800,720,1,

%T 161406,141246,91560,47040,15120,5040,1,4747010,3444756,2162664,

%U 1134000,493920,141120,40320,1,194342910,110729310,61286400,32253480,14605920,5594400

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

%C 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.

%H Joel B. Lewis, <a href="/A222864/b222864.txt">Rows n = 1..20 of triangle, flattened</a>

%H J. B. Lewis and Y. X. Zhang, <a href="http://arxiv.org/abs/1106.5480">Enumeration of Graded (3+1)-Avoiding Posets</a>, J. Combin. Theory Ser. A 120 (2013), no. 6, 1305-1327.

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

%e 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.

%Y 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.

%K nonn,tabl

%O 1,3

%A _Joel B. Lewis_, Mar 07 2013