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

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, 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, 1, 0, 13, 0, 36, 0, 0, 3, 0, 23, 0, 24, 0, 0, 3, 0, 12, 0

Offset: 1

Author

Roger Ford, Dec 09 2015

Keywords

• nonn
• tabf

Comments

A positive arch is defined as a top arch that starts at an odd-numbered vertex and ends at a higher even-numbered vertex.

For each value of n there is a square array with n^2 elements.

Rows are in order of decreasing number of components.

The sum of all the elements in each square array(n) = Catalan numbers C(n) A000108.

The sum of columns for array(n) = Semimeander components row(n) A046726.

The sum of the rows for array(n) = Narayana numbers T(n,k) A001263.

All semimeander solutions (k=1) for array n have positive arches = floor((n+2)/2).

Examples

			For n=3:                                   /\          /\
               /\               /\        /  \        //\\
              /  \             /  \      /    \      //  \\
  /\ /\ /\   / /\ \  /\   /\  / /\ \    //\  /\\    // /\ \\
  \ \\// /   \ \ \/ / /   \ \ \/ / /    \\ \/ //    \\ \/ //
   \ \/ /     \ \  / /     \ \  / /      \\  //      \\  //
    \  /       \ \/ /       \ \/ /        \\//        \\//
     \/         \  /         \  /          \/          \/
                 \/           \/
  p=3,k=2     p=2,k=1      p=2,k=1      p=1,k=2     p=2,k=3.
.
n=3  p\k 3  2  1   n=9  p\k 9  8  7  6  5  4  3  2  1
      1: 0  1  0         1: 0  0  0  0  1  0  0  0  0
      2: 1  0  2         2: 0  0  0  4  0 32  0  0  0
      3: 0  1  0         3: 0  0  6  0 78  0 252 0  0
                         4: 0  4  0 72  0 446 0 654 0
                         5: 1  0 29  0 280 0 950 0 504
                         6: 0  4  0 72  0 446 0 654 0
                         7: 0  0  6  0 78  0 252 0  0
                         8: 0  0  0  4  0 32  0  0  0
                         9: 0  0  0  0  1  0  0  0  0
		

Crossrefs

Cf. A000108, A001263, A046726.