A005271 Number of perfect matchings in n-cube.

1, 2, 9, 272, 589185, 16332454526976, 391689748492473664721077609089

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice

Comments

The matchings contain 2^n / 2 = 2^(n-1) edges.

a(6) was first found by D. H. Wiedemann, unpublished (see Clark et al., Skupien).

Also the number of minimum edge covers and minimum clique coverings in the n-hypercube graph. - Eric W. Weisstein, Dec 24 2017

Examples

			G.f. = x + 2*x^2 + 9*x^3 + 272*x^4 + 589185*x^5 + 16332454526976*x^6 + ...
		

References

• L. H. Clark, J. C. George and T. D. Porter, On the number of 1-factors in the n-cube, Congress. Numer., 127 (1997), 67-69.
• J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 18).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• M. K. Chari and M. Joswig, Complexes of discrete Morse functions, Disc. Math. 302 (2005), 39-51.
• D. Deford, Seating rearrangements on arbitrary graphs, Involve 7(6): 787-805 (2014); See Table 3.
• N. Graham and F. Harary, The number of perfect matchings in a hypercube, Appl. Math. Lett., 1 (1988), 45-48.
• N. Graham and F. Harary, The number of perfect matchings in a hypercube, Appl. Math. Lett. 1.1 (1988), 45-48. (Annotated scanned copy)
• Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden.
• Patric R. J. Östergård and V. H. Pettersson, Enumerating Perfect Matchings in n-Cubes, Order, November 2013, Volume 30, Issue 3, pp 821-835.
• Ville Pettersson, Graph Algorithms for Constructing and Enumerating Cycles and Related Structures, Preprint 2015.
• J. Propp, Twenty open problems in enumeration of matchings, arXiv:math/9801061 [math.CO], 1998-1999.
• J. Propp, Updated article
• J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
• H. Sachs and B. Alspach, Problem 298: How many perfect matchings does the graph of the n-cube have?, Discrete Math., 191 (1998), 251-252. [From _N. J. A. Sloane_, Feb 18 2012]
• Eric Weisstein's World of Mathematics, Hypercube Graph
• Eric Weisstein's World of Mathematics, Matching
• Eric Weisstein's World of Mathematics, Maximum Independent Edge Set
• Eric Weisstein's World of Mathematics, Minimum Clique Covering
• Eric Weisstein's World of Mathematics, Minimum Edge Cover
• Eric Weisstein's World of Mathematics, Perfect Matching

Crossrefs

Cf. A220904, A112311.

For all matchings see A045310.

Extensions

a(6) from Per H. Lundow, Jul 15 1996

a(7) from N. J. A. Sloane, Jan 01 2013