A331499 T(n,k) is the number of nonintersecting arch configurations with n arches on a horizontal axis that have a starting parent arch configuration of k arches.

1, 1, 1, 2, 1, 2, 4, 3, 2, 5, 10, 6, 7, 5, 14, 24, 18, 15, 19, 14, 42, 66, 42, 48, 43, 56, 42, 132, 174, 130, 116, 143, 132, 174, 132, 429, 504, 332, 374, 358, 451, 423, 561, 429, 1430, 1406, 1048, 974, 1182, 1163, 1475, 1397, 1859, 1430, 4862

Offset: 1

Author

Roger Ford, Feb 16 2020

Keywords

• nonn
• tabl

Comments

A parent arch configuration has exactly one exterior arch and can be derived from an initial arch configuration with more than one exterior arch by using the algorithm for decreasing arches explained below.

When concentric arches are placed below the different arch configurations, the number of components (loops formed) in each configuration will remain the same as the number components in that arch configuration's parent arch.

Algorithm: To determine the parent arch configuration, start with all the nonintersecting arch configurations with n arches and more than one exterior arch. For each starting individual arch configuration, reduce the number of arches until there is exactly one exterior arch. Join the start of the first arch with the end of the last arch. This will eliminate one arch and shrink the number of arches from n to n-1. Continue this process until there is one exterior arch.

/\ n=5

/ \ /\ n=5

/ /\\ /\ / \

--> /\//\//\\\ <-- --> //\\//\/\\ <--

/\

--> /\/\//\\ <-- --> /\/\/\/\ <--

/\ /\

--> //\\/\ <-- / \

--> /\/\ <-- //\/\\ parent arch k = 3

/\ parent arch k = 1.

Examples

			Triangle T(n,k) begins:
  n\k    1    2    3    4    5    6    7    8    9   10    11
   1:    1
   2:    1    1
   3:    2    1    2
   4:    4    3    2    5
   5:   10    6    7    5   14
   6:   24   18   15   19   14   42
   7:   66   42   48   43   56   42  132
   8:  174  130  116  143  132  174  132  429
   9:  504  332  374  358  451  423  561  429 1430
  10: 1406 1048  974 1182 1163 1475 1397 1859 1430 4862
  11: 4210 2836 3166 3152 3906 3897 4950 4719 6292 4862 16796
		

Crossrefs

Cf. A000108, A000682.

Formula

The sum of row n = A000108(n).

T(n,1) = A000682(n).

T(n,n) = A000108(n-1).

T(n,n-1) = A000108(n-2).

T(n,n-2) = A000108(n-2) + A000108(n-3) (conjectured).