A318243 Triangle read by rows giving the sum of the number of k-matchings of the graphs obtained by deleting one edge and its two vertices from the ladder graph L_n = P_2 X P_n in all possible ways.

1, 4, 4, 7, 22, 9, 10, 58, 78, 20, 13, 112, 282, 224, 40, 16, 184, 702, 1052, 570, 78, 19, 274, 1419, 3260, 3335, 1338, 147, 22, 382, 2514, 7928, 12520, 9462, 2968, 272, 25, 508, 4068, 16460, 35955, 42108, 24766, 6312, 495, 28, 652, 6162, 30584, 86330, 140586, 128352, 60976, 12996, 890, 31, 814, 8877, 52352

Offset: 1

Author

Donovan Young, Aug 22 2018

Keywords

• nonn
• tabl

Comments

T(n,k) is useful for computing the number of configurations of n indistinguishable pairs placed on the vertices of P_2 X P_n such that only one such pair is joined by an edge. The g.f. given below can be proven using the calculus of the rook polynomial associated with A046741.

Examples

			The first few rows of T(n,k) are
   1;
   4,   4;
   7,  22,   9;
  10,  58,  78,  20;
  13, 112, 282, 224,  40;
For n = 2, the four ways of deleting an edge and its vertices from P_2 X P_2 all yield a graph with two vertices joined by an edge. This graph has 1 0-matching and 1 1-matching, thus T(2,k) = 4, 4.
		

Links

• D. Young, The Number of Domino Matchings in the Game of Memory, Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.1.
• Donovan Young, Generating Functions for Domino Matchings in the 2 * k Game of Memory, arXiv:1905.13165 [math.CO], 2019. Also in J. Int. Seq., Vol. 22 (2019), Article 19.8.7.

Crossrefs

Cf. A046741, A318244, A318267, A318268, A318269, A318270.

Programs

Mathematica

CoefficientList[Normal[Series[(1 + 2*t*z + 2*z/(1-t*z)^2)*(1-t*z)^2/(1 - z - 2*t*z - t*z^2 + t^3*z^3)^2,{z,0,10}]],{z,t}]//MatrixForm

Formula

G.f.: (1 + 2*t*z + 2*z/(1-t*z)^2)*(1-t*z)^2/(1 - z - 2*t*z - t*z^2 + t^3*z^3)^2.