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A265507 A pyramid T(n,p,k) of square arrays read by rows relating semimeanders(n), positive arches(p) and components(k).

%I A265507 #19 Feb 07 2016 20:06:31
%S A265507 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 A265507 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 A265507 1,0,13,0,36,0,0,3,0,23,0,24,0,0,3,0,12,0
%N A265507 A pyramid T(n,p,k) of square arrays read by rows relating semimeanders(n), positive arches(p) and components(k).
%C A265507 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 A265507 For each value of n there is a square array with n^2 elements.
%C A265507 Rows are in order of decreasing number of components.
%C A265507 The sum of all the elements in each square array(n) = Catalan numbers C(n) A000108.
%C A265507 The sum of columns for array(n) = Semimeander components row(n) A046726.
%C A265507 The sum of the rows for array(n) = Narayana numbers T(n,k) A001263.
%C A265507 All semimeander solutions (k=1) for array n have positive arches = floor((n+2)/2).
%e A265507 For n=3:                                   /\          /\
%e A265507                /\               /\        /  \        //\\
%e A265507               /  \             /  \      /    \      //  \\
%e A265507   /\ /\ /\   / /\ \  /\   /\  / /\ \    //\  /\\    // /\ \\
%e A265507   \ \\// /   \ \ \/ / /   \ \ \/ / /    \\ \/ //    \\ \/ //
%e A265507    \ \/ /     \ \  / /     \ \  / /      \\  //      \\  //
%e A265507     \  /       \ \/ /       \ \/ /        \\//        \\//
%e A265507      \/         \  /         \  /          \/          \/
%e A265507                  \/           \/
%e A265507   p=3,k=2     p=2,k=1      p=2,k=1      p=1,k=2     p=2,k=3.
%e A265507 .
%e A265507 n=3  p\k 3  2  1   n=9  p\k 9  8  7  6  5  4  3  2  1
%e A265507       1: 0  1  0         1: 0  0  0  0  1  0  0  0  0
%e A265507       2: 1  0  2         2: 0  0  0  4  0 32  0  0  0
%e A265507       3: 0  1  0         3: 0  0  6  0 78  0 252 0  0
%e A265507                          4: 0  4  0 72  0 446 0 654 0
%e A265507                          5: 1  0 29  0 280 0 950 0 504
%e A265507                          6: 0  4  0 72  0 446 0 654 0
%e A265507                          7: 0  0  6  0 78  0 252 0  0
%e A265507                          8: 0  0  0  4  0 32  0  0  0
%e A265507                          9: 0  0  0  0  1  0  0  0  0
%Y A265507 Cf. A000108, A001263, A046726.
%K A265507 nonn,tabf
%O A265507 1,11
%A A265507 _Roger Ford_, Dec 09 2015