A060972 Triangle giving numbers of closed plane meanders.

1, 1, 1, 1, 6, 1, 1, 20, 20, 1, 1, 50, 160, 50, 1, 1, 105, 808, 808, 105, 1, 1, 196, 3066, 7294, 3066, 196, 1, 1, 336, 9552, 45588, 45588, 9552, 336, 1, 1, 540, 25740, 220362, 440172, 220362, 25740, 540, 1, 1, 825, 62040, 879840, 3133724, 3133724, 879840, 62040, 825, 1, 1, 1210, 136851, 3028454, 17752636, 31586346, 17752636, 3028454, 136851, 1210, 1

Offset: 1

Author

F. Chapoton, May 09 2001; extended to 14 rows, Jul 31 2011

Keywords

• nonn
• tabl

Comments

a(n) counts closed plane meanders according to the number of white regions when regions are colored black and white alternatively. So the sum of each row is given by A005315. The outer columns consist of 1's. The next-to-outer columns are given by A002415.

This is also the number of arches above the x-axis going from an odd vertex to a higher even vertex(p) for closed plane meanders(M) with n arches. By symmetry, these same subsets exist for arches below the x-axis. For each meander solution, the total arches for the top and bottom that go from an odd vertex to a higher even vertex is n+1.

Example: M(n,p): M(3,1)=1 [(top 16,23,45; bottom 12,34,56)], M(3,2)=6 [(top 14,23,56; bottom 16,25,34)(top 16,25,34; bottom 14,23,56) (top 12,36,45; bottom 16,25,34) (top 16,25,34; bottom 12,36,45) (top 12,36,45; bottom 14,23,56)(top 14,23,56; bottom 12,36,45)] M(3,3)=1 [(top 12,34,56; bottom 16,23,45)]. - Roger Ford, Sep 29 2014

Examples

			Triangle begins:
1;
1, 1;
1, 6, 1;
1, 20, 20, 1;
1, 50, 160, 50, 1;
1, 105, 808, 808, 105, 1; ...
		

Links

• Joerg Arndt and Andrew Howroyd, Table of n, a(n) for n = 1..210 (first 105 terms from Joerg Arndt)
• Reinhard O. W. Franz, and Berton A. Earnshaw, A constructive enumeration of meanders, Ann. Comb. 6 (2002), no. 1, 7-17. [Table 1 gives first 14 rows]

Crossrefs

Row sums give A005315, diagonals give A002415.