A362308 Triangle read by rows. Number of perfect matchings by number of connected components.

1, 0, 1, 0, 2, 1, 0, 10, 4, 1, 0, 74, 24, 6, 1, 0, 706, 188, 42, 8, 1, 0, 8162, 1808, 350, 64, 10, 1, 0, 110410, 20628, 3426, 568, 90, 12, 1, 0, 1708394, 273064, 38886, 5696, 850, 120, 14, 1

Offset: 0

Author

Peter Luschny, Apr 15 2023

Keywords

• nonn
• tabl
• more

Comments

The exact definition is given in Sokal and Zeng. See section 4.4 and theorem 4.6.

Examples

			Table T(n, k) begins:
  [0] 1;
  [1] 0,       1;
  [2] 0,       2,      1;
  [3] 0,      10,      4,     1;
  [4] 0,      74,     24,     6,    1;
  [5] 0,     706,    188,    42,    8,   1;
  [6] 0,    8162,   1808,   350,   64,  10,   1;
  [7] 0,  110410,  20628,  3426,  568,  90,  12,  1;
  [8] 0, 1708394, 273064, 38886, 5696, 850, 120, 14, 1;
		

Links

• Alan D. Sokal and Jiang Zeng, Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions, Advances in Applied Mathematics, Volume 138, 2022. Table on p. 91.
• Wikipedia, Perfect matching.

Crossrefs

Cf. A001147 (row sums), A000698 (indecomposable perfect matchings), A177797.

T(n,0) = A000007(n), T(n,1) = A000698(n) assuming offset 1.

Formula

T(n, k) = T(n, k-1) - T(n-1, k-2) - (2*n - k - 1)/(k - 1) * T(n - 1, k - 1) for k > 1. - Detlef Meya, Dec 21 2023