A384121 Array read by antidiagonals: T(n,m) is the number of dominating sets in the n X m rook complement graph.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 39, 39, 1, 1, 1, 1, 183, 421, 183, 1, 1, 1, 1, 833, 3825, 3825, 833, 1, 1, 1, 1, 3629, 32047, 64727, 32047, 3629, 1, 1, 1, 1, 15291, 260355, 1046425, 1046425, 260355, 15291, 1, 1, 1, 1, 63051, 2092909, 16771879, 33548731, 16771879, 2092909, 63051, 1, 1

Offset

0,13

Comments

Non-dominating sets are just those that are contained in the union of a single row and column minus the intersecting vertex.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Eric Weisstein's World of Mathematics, Dominating Set.

Eric Weisstein's World of Mathematics, Rook Complement Graph.

Formula

T(n,m) = 2^(n*m) - n*(2^m-2) - m*(2^n-2) + n*m - n*m*(2^(m-1)-1)*(2^(n-1)-1) + n*(n-1)*m*(m-1)/2 - 1 for n > 1, m > 1.

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

Example

Array begins:
===============================================================
n\m | 0 1     2       3         4           5             6 ...
----+----------------------------------------------------------
  0 | 1 1     1       1         1           1             1 ...
  1 | 1 1     1       1         1           1             1 ...
  2 | 1 1     9      39       183         833          3629 ...
  3 | 1 1    39     421      3825       32047        260355 ...
  4 | 1 1   183    3825     64727     1046425      16771879 ...
  5 | 1 1   833   32047   1046425    33548731    1073727713 ...
  6 | 1 1  3629  260355  16771879  1073727713   68719441881 ...
  7 | 1 1 15291 2092909 268422785 34359704907 4398046428559 ...
  ...

Prog

(PARI) T(n, m) = if(n<=1 || m<=1, 1, 2^(n*m) - n*(2^m-2) - m*(2^n-2) + n*m - n*m*(2^(m-1)-1)*(2^(n-1)-1) + n*(n-1)*m*(m-1)/2 - 1)

Crossrefs

Main diagonal is A292073.

Columns 0 and 1 are A000012.

Column 2 is A287063, n > 1.

Cf. A384120 (independent sets), A384122, A384123.

Sequence in context: A087968 A340365 A110483 * A348734 A010164 A006084

Adjacent sequences: A384118 A384119 A384120 * A384122 A384123 A384124

Keyword

nonn,tabl

Author

Andrew Howroyd, May 20 2025

Status

approved