A368831 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with cardinality k of the n X n rook graph (n >= 0, 0 <= k <= n^2).

1, 0, 1, 0, 0, 6, 4, 1, 0, 0, 0, 48, 117, 126, 84, 36, 9, 1, 0, 0, 0, 0, 488, 2640, 6712, 10864, 12726, 11424, 8008, 4368, 1820, 560, 120, 16, 1, 0, 0, 0, 0, 0, 6130, 58300, 269500, 808325, 1778875, 3075160, 4349400, 5154900, 5186300, 4454400, 3268360, 2042950, 1081575, 480700, 177100, 53130, 12650, 2300, 300, 25, 1

Offset: 0

Author

Stephan Mertens, Jan 07 2024

Keywords

• nonn
• tabf

Comments

The entries in row n are the coefficients of the domination polynomial of the n X n rook graph.

Sum of entries in row n = A287065 = main diagonal of A287274.

Number of minimum dominating sets T(n,n) = A248744(n).

Examples

			Triangle begins: (first 5 rows)
  1;
  0, 1;
  0, 0, 6,  4,   1;
  0, 0, 0, 48, 117,  126,   84,    36,     9,     1;
  0, 0, 0,  0, 488, 2640, 6712, 10864, 12726, 11424, 8008, 4368, 1820, 560, 120, 16, 1;
  ...
		

References

• John J. Watkins, Across the Board: The Mathematics of Chessboard Problems, Princeton University Press, 2004, chapter 7.

Links

• Alois P. Heinz, Rows n = 0..32, flattened
• Stephan Mertens, Domination Polynomial of the Rook Graph, Journal of Integer Sequences 27 (2024), Article 24.3.7; arXiv:2401.00716 [math.CO], 2024.
• Eric Weisstein's World of Mathematics, Dominating Set.
• Eric Weisstein's World of Mathematics, Rook Graph.

Crossrefs

Cf. A000290, A083374, A287065 (row sums), A287274, A248744 (leading diagonal).

Programs

Mathematica

R[n_, m_] := CoefficientList[((x + 1)^n - 1)^m - (-1)^m*Sum[Binomial[m, k]*(-1)^k*((1 + x)^k - 1)^n, {k, 0, m - 1}], x];
Flatten[Table[R[n,n],{n,1,5}]]

Formula

G.f.: ((x+1)^n - 1)^m - (-1)^m * Sum_{k=0..m-1} binomial(m,k)*(-1)^k*((1+x)^k - 1)^n (for the rectangular n X m rook graph).

T(n,n) = 2*n^n - n!.