Stephan Mertens - cp's OEIS frontend

A375603 Array read by antidiagonals: T(m,n) = domination number of the stacked prism graph C_m X P_n.

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 4, 3, 3, 2, 3, 4, 4, 4, 4, 4, 3, 3, 4, 5, 5, 6, 5, 4, 3, 3, 5, 6, 6, 7, 6, 6, 4, 3, 4, 5, 7, 7, 8, 8, 7, 6, 5, 4, 4, 6, 7, 8, 9, 9, 9, 8, 7, 6, 4, 4, 6, 8, 9, 10, 10, 11, 10, 10, 8, 6, 4, 5, 7, 9, 10, 11, 12, 12, 12, 12, 10, 9, 6, 5

Offset: 1

Stephan Mertens, Aug 20 2024

			Table starts:
====================================
m\n |   1   2   3    4    5    6 ...
----|-------------------------------
  1 |   1   1   1    2    2    2 ...
  2 |   1   2   2    3    3    4 ...
  3 |   1   2   3    4    4    5 ...
  4 |   2   2   3    4    5    6 ...
  5 |   2   3   4    6    7    8 ...
  6 |   2   4   5    6    8    9 ...
 ...

Stephan Mertens, Table of n, a(n) for n = 1..325
Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], Aug 2024.
Eric Weisstein's World of Mathematics, Domination Number.
Eric Weisstein's World of Mathematics, Stacked Prism Graph.

Main diagonal is A375601.

Cf. A286514, A350823, A375566.

A375601 a(n) = domination number of the stacked prism graph C_n X P_n.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 12, 16, 20, 24, 28, 34, 40, 45, 51, 58, 65, 73, 81, 88, 98, 107, 117, 127

Offset: 1

Stephan Mertens, Aug 20 2024

Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], Aug 2024.
Eric Weisstein's World of Mathematics, Domination Number.
Eric Weisstein's World of Mathematics, Stacked Prism Graph.

Main diagonal of A375603.

Cf. A375569.

A375566 Array read by antidiagonals: T(m,n) = number of minimum dominating sets in the stacked prism graph C_m X P_n.

Original entry on oeis.org

1, 2, 2, 1, 6, 3, 4, 3, 9, 6, 3, 12, 34, 4, 5, 1, 2, 123, 4, 10, 3, 8, 17, 3, 16, 5, 51, 14, 4, 2, 18, 28, 290, 18, 14, 8, 1, 20, 93, 76, 320, 6, 63, 4, 3, 13, 2, 438, 164, 265, 171, 14, 4, 18, 25, 5, 24, 3, 396, 255, 36, 91, 24, 9, 120, 11, 1, 2, 27, 904, 250, 6, 1526, 60, 2052, 25, 22, 3

Offset: 1

Stephan Mertens, Aug 19 2024

			Table starts:
====================================
m\n |   1   2   3    4    5    6 ...
----+-------------------------------
  1 |   1   2   1    4    3    1 ...
  2 |   2   6   3   12    2   17 ...
  3 |   3   9  34  123    3   18 ...
  4 |   6   4   4   16   28   76 ...
  5 |   5  10   5  290  320  265 ...
 ...

Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], Aug 2024.
Eric Weisstein's World of Mathematics, Minimum Dominating Set.
Eric Weisstein's World of Mathematics, Stacked Prism Graph.

Main diagonal is A375569.
Rows 1..2 are A347633, A347558.
Column 1 is A347538, column 2 is essentially A347634.
Cf. A286514, A350820, A375603.

A375569 a(n) = number of minimum dominating sets in the stacked prism graph C_n X P_n.

Original entry on oeis.org

1, 6, 34, 16, 320, 36, 56, 5556, 20196, 32210, 88, 121428, 388284, 224, 1489960, 12800, 251464, 2304, 36784, 73062090, 29787744, 738959760, 73600, 884736

Offset: 1

Stephan Mertens, Aug 19 2024

Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], Aug 2024.
Eric Weisstein's World of Mathematics, Minimum Dominating Set.
Eric Weisstein's World of Mathematics, Stacked Prism Graph.

Main diagonal of A375566.

Cf. A286914, A347557, A347632.

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).

Original entry on oeis.org

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

Stephan Mertens, Jan 07 2024

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).

			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;
  ...

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

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.

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

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}]]

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!.

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User: Stephan Mertens

A375603 Array read by antidiagonals: T(m,n) = domination number of the stacked prism graph C_m X P_n.

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A375601 a(n) = domination number of the stacked prism graph C_n X P_n.

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A375566 Array read by antidiagonals: T(m,n) = number of minimum dominating sets in the stacked prism graph C_m X P_n.

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A375569 a(n) = number of minimum dominating sets in the stacked prism graph C_n X P_n.

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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).

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