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%I #18 Sep 03 2021 06:55:13
%S 1,3,9,21,48,102,213,421,819,1542,2854,5172,9240,16233,28182,48288,
%T 81862,137295,228153,375658,613554,994155,1599309,2554932,4055406,
%U 6397160,10032907,15647277,24275455,37471066,57562533,88018488,133996590,203126712,306671525,461184246,690935892,1031379271
%N a(n) is the number of pairs (x,y) where x is plane partition of n+1 and y is a plane partition of n and x covers y.
%C x = (x_1_1 .. x_1_u1)(x_2_1 .. x_2_u2) .. (x_k_1 .. x_k_uk) y = (y_1_1 .. y_1_v1)(y_2_1 .. y-2_v2) .. (y_m_1 .. y_m_vm) x covers y iff ui >= vi, k >= m, x_i_j >= y_i_j, or, the 3-dimensional Ferrers plot of y falls within that of x.
%C The analog for ordinary partitions and 2D-Ferrers plots gives A000070.
%H Suresh Govindarajan, <a href="/A090984/b090984.txt">Table of n, a(n) for n = 0..40</a>
%t coversplaneQ[parent_?planepartitionQ, child_?planepartitionQ] := Block[{dif=Length[parent]-Length[child], p=Length/@ parent, c=PadRight[Length/@ child, Length[parent], 0]}, And[dif>=0, Min[p-c]>=0, Min[parent-MapThread[PadRight[ #1, #2, 0]&, { PadRight[child, Length[parent], {{0}}], p}]]>=0]]; Table[Count[Outer[coversplaneQ, planepartitions[k], planepartitions[k-1], 1], True, -1], {k, 12}]
%Y Cf. A000070, A090539.
%K nonn
%O 0,2
%A _Wouter Meeussen_, Feb 28 2004
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