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%I #19 Feb 07 2016 20:06:31
%S 1,1,0,0,1,0,1,0,1,0,2,0,1,0,0,1,0,0,1,0,5,0,0,2,0,4,0,0,1,0,0,0,1,0,
%T 0,0,2,0,8,0,1,0,9,0,10,0,2,0,8,0,0,0,1,0,0,0,0,1,0,0,0,0,2,0,13,0,0,
%U 1,0,13,0,36,0,0,3,0,23,0,24,0,0,3,0,12,0
%N A pyramid T(n,p,k) of square arrays read by rows relating semimeanders(n), positive arches(p) and components(k).
%C A positive arch is defined as a top arch that starts at an odd-numbered vertex and ends at a higher even-numbered vertex.
%C For each value of n there is a square array with n^2 elements.
%C Rows are in order of decreasing number of components.
%C The sum of all the elements in each square array(n) = Catalan numbers C(n) A000108.
%C The sum of columns for array(n) = Semimeander components row(n) A046726.
%C The sum of the rows for array(n) = Narayana numbers T(n,k) A001263.
%C All semimeander solutions (k=1) for array n have positive arches = floor((n+2)/2).
%e For n=3: /\ /\
%e /\ /\ / \ //\\
%e / \ / \ / \ // \\
%e /\ /\ /\ / /\ \ /\ /\ / /\ \ //\ /\\ // /\ \\
%e \ \\// / \ \ \/ / / \ \ \/ / / \\ \/ // \\ \/ //
%e \ \/ / \ \ / / \ \ / / \\ // \\ //
%e \ / \ \/ / \ \/ / \\// \\//
%e \/ \ / \ / \/ \/
%e \/ \/
%e p=3,k=2 p=2,k=1 p=2,k=1 p=1,k=2 p=2,k=3.
%e .
%e n=3 p\k 3 2 1 n=9 p\k 9 8 7 6 5 4 3 2 1
%e 1: 0 1 0 1: 0 0 0 0 1 0 0 0 0
%e 2: 1 0 2 2: 0 0 0 4 0 32 0 0 0
%e 3: 0 1 0 3: 0 0 6 0 78 0 252 0 0
%e 4: 0 4 0 72 0 446 0 654 0
%e 5: 1 0 29 0 280 0 950 0 504
%e 6: 0 4 0 72 0 446 0 654 0
%e 7: 0 0 6 0 78 0 252 0 0
%e 8: 0 0 0 4 0 32 0 0 0
%e 9: 0 0 0 0 1 0 0 0 0
%Y Cf. A000108, A001263, A046726.
%K nonn,tabf
%O 1,11
%A _Roger Ford_, Dec 09 2015
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