A303209 -id:A303209 - cp's OEIS frontend

A303212 Number of minimum total dominating sets in the n X n rook complement graph.

0, 1, 6, 96, 600, 2400, 7350, 18816, 42336, 86400, 163350, 290400, 490776, 794976, 1242150, 1881600, 2774400, 3995136, 5633766, 7797600, 10613400, 14229600, 18818646, 24579456, 31740000, 40560000, 51333750, 64393056, 80110296, 98901600

Offset: 1

Eric W. Weisstein, Apr 19 2018

For n > 2, the minimum total dominating sets are any three vertices such that no two are in the same row or column. - Andrew Howroyd, Apr 20 2018

Essentially the same as A179058(n), differing only for n=2. - Eric W. Weisstein, Dec 06 2023

Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rook Complement Graph.
Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A179058, A303209, A347922.

Table[If[n == 2, 1, 6 Binomial[n, 3]^2], {n, 20}]
Join[{0, 1}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 6, 96, 600, 2400, 7350}, {3, 20}]]
CoefficientList[Series[x (-1 + x - 75 x^2 - 19 x^3 - 41 x^4 + 21 x^5 - 7 x^6 + x^7)/(-1 + x)^7, {x, 0, 20}], x]

a(n) = if(n<3, n==2, 6*binomial(n,3)^2) \\ Andrew Howroyd, Apr 20 2018

concat(0, Vec(x^2*(1 - x + 75*x^2 + 19*x^3 + 41*x^4 - 21*x^5 + 7*x^6 - x^7) / (1 - x)^7 + O(x^60))) \\ Colin Barker, Apr 20 2018

a(n) = A179058(n) for n > 2. - Andrew Howroyd, Apr 20 2018
From Colin Barker, Apr 20 2018: (Start)
G.f.: x^2*(1 - x + 75*x^2 + 19*x^3 + 41*x^4 - 21*x^5 + 7*x^6 - x^7) / (1 - x)^7.
a(n) = n^2*(2 - 3*n + n^2)^2 / 6 for n > 2.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 9.
(End)

a(6)-a(30) from Andrew Howroyd, Apr 20 2018

A347922 Number of minimal total dominating sets in the n X n rook complement graph.

Original entry on oeis.org

0, 1, 51, 492, 2500, 8925, 25431, 61936, 134352, 266625, 493075, 861036, 1433796, 2293837, 3546375, 5323200, 7786816, 11134881, 15604947, 21479500, 29091300, 38829021, 51143191, 66552432, 85650000, 109110625, 137697651, 172270476, 213792292, 263338125

Offset: 1

Eric W. Weisstein, Sep 19 2021

From Andrew Howroyd, Jan 19 2022: (Start)
The vertex sets which are not totally dominating are just those that are contained in the union of a single row and column. Minimal total dominating sets are:
  - any three vertices such that no two are in the same row or column,
  - two vertices in each of two rows/columns. (End)

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Minimal Total Dominating Set
Eric Weisstein's World of Mathematics, Rook Complement Graph
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A292074, A303209, A303212.

Table[(n - 1)^2 n^2 (5 n^2 - 11 n + 5)/12, {n, 20}] (* Eric W. Weisstein, May 11 2024 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 1, 51, 492, 2500, 8925, 25431}, 20] (* Eric W. Weisstein, May 11 2024 *)
CoefficientList[Series[-x (1 + 44 x + 156 x^2 + 92 x^3 + 7 x^4)/(-1 + x)^7, {x, 0, 20}], x] (* Eric W. Weisstein, May 11 2024 *)

a(n) = (5*n^2 - 11*n + 5)*n^2*(n-1)^2/12 \\ Andrew Howroyd, Jan 19 2022

From Andrew Howroyd, Jan 19 2022: (Start)
a(n) = 6*binomial(n,3)^2 + 2*binomial(n,2)^3 - binomial(n,2)^2.
a(n) = (5*n^2 - 11*n + 5)*n^2*(n-1)^2/12.
G.f.: x*(1 + 44*x + 156*x^2 + 92*x^3 + 7*x^4)/(1 - x)^7.
(End)

Terms a(6) and beyond from Andrew Howroyd, Jan 19 2022

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A303212 Number of minimum total dominating sets in the n X n rook complement graph.

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A347922 Number of minimal total dominating sets in the n X n rook complement graph.

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