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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

%I #18 May 26 2013 15:01:09

%S 1,1,2,1,12,6,1,86,84,24,1,840,1110,480,120,1,11642,16620,9120,3240,

%T 720,1,227892,300846,185640,82320,25200,5040,1,6285806,6810804,

%U 4299624,2142000,816480,221760,40320,1,243593040,199239270,117205200,60890760,26157600

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

%C 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.

%H Joel B. Lewis, <a href="/A222866/b222866.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.

%Y 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.

%K nonn,tabl

%O 1,3

%A _Joel B. Lewis_, Mar 07 2013

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