A359993 Array read by antidiagonals: T(m,n) is the number of connected spanning subgraphs in the grid graph P_m X P_n.

1, 1, 1, 1, 5, 1, 1, 23, 23, 1, 1, 105, 431, 105, 1, 1, 479, 7857, 7857, 479, 1, 1, 2185, 142625, 555195, 142625, 2185, 1, 1, 9967, 2587279, 38757695, 38757695, 2587279, 9967, 1, 1, 45465, 46929343, 2698167665, 10286937043, 2698167665, 46929343, 45465, 1

Offset: 1

Author

Andrew Howroyd, Jan 28 2023

Keywords

• nonn
• tabl

Comments

Also T(m,n) except when m = n = 0 is the number of connected edge covers in the m X n grid graph.

Examples

			Table starts:
=================================================================
m\n| 1    2       3          4             5                6
---+-------------------------------------------------------------
1  | 1    1       1          1             1                1 ...
2  | 1    5      23        105           479             2185 ...
3  | 1   23     431       7857        142625          2587279 ...
4  | 1  105    7857     555195      38757695       2698167665 ...
5  | 1  479  142625   38757695   10286937043    2711895924889 ...
6  | 1 2185 2587279 2698167665 2711895924889 2692324030864335 ...
   ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..465
• Eric Weisstein's World of Mathematics, Grid Graph

Crossrefs

Rows 1..4 are A000012, A107839(n-1), A158453, A359991.

Main diagonal is A359992.

Cf. A116469 (spanning trees), A287151 (connected induced subgraphs), A286912 (edge covers), A359990 (edge cuts), A360194 (spanning forests).

Formula

T(m,n) = T(n,m).