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A005271 Number of perfect matchings in n-cube.
(Formerly M1955)
1, 2, 9, 272, 589185, 16332454526976, 391689748492473664721077609089 (list; graph; refs; listen; history; text; internal format)



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a(6) was first found by D. H. Wiedemann, unpublished (see Clark et al., Skupien).


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).


Table of n, a(n) for n=1..7.

M. K. Chari and M. Joswig, Complexes of discrete Morse functions, Disc. Math. 302 (2005), 39-51.

D. Deford, Seating rearrangements on arbitrary graphs, 2013; See Table 3.

N. Graham and F. Harary, The number of perfect matchings in a hypercube, Appl. Math. Lett., 1 (1988), 45-48.

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.

P. R. J. Ostergard 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, 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, Perfect Matching


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


Cf. A220904, A112311.

For all matchings see A045310.

Sequence in context: A122894 A042675 A015177 * A258668 A012938 A013093

Adjacent sequences:  A005268 A005269 A005270 * A005272 A005273 A005274




N. J. A. Sloane.


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

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



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Last modified May 29 22:45 EDT 2017. Contains 287257 sequences.