0,3

Number of graphs on n labeled nodes; also number of outcomes of labeled n-team round-robin tournaments.

Number of perfect matchings of order n Aztec diamond [see Speyer]

Number of Gelfand-Zeitlin patterns with bottom row [1,2,3,...,n]. [Zeilberger]

For n >= 1 a(n) is the size of the Sylow 2-subgroup of the Chevalley group A_n(2) (sequence A002884) - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001

Comment from Jim Propp (propp(AT)math.wisc.edu): a(n) is the number of ways to tile the region

.........o-----o

.........|.....|

......o--o.....o--o

......|...........|

...o--o...........o--o

...|.................|

o--o.................o--o

|.......................|

|.......................|

|.......................|

o--o.................o--o

...|.................|

...o--o...........o--o

......|...........|

......o--o.....o--o

.........|.....|

.........o-----o

(top-to-bottom distance = 2n) with dominoes like either of

o--o o-----o

|..| |.....|

|..| o-----o

|..|

o--o

The number of domino tilings in A006253, A004003, A006125 is the number of perfect matchings in the relevant graphs. There are results of Jockusch and Ciucu that if a planar graph has a rotational symmetry then the number of perfect matchings is a square or twice a square - this applies to these 3 sequences. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001

Let M_n denotes the n X n matrix with M_n(i,j)=binomial(2i,j); then det(M_n)=a(n+1). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 21 2002

Smallest power of 2 which can be expressed as the product of n distinct numbers (powers of 2), e.g. a(4) = 1024 = 2*4*8*16. Also smallest number which can be expressed as the product of n distinct powers. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 10 2002

The number of binary relations that are both reflexive and symmetric on an n-element set. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005

To win a game, you must flip n+1 heads in a row, where n is the total number of tails flipped so far. Then the probability of winning for the first time after n tails is A005329 / A006125 . The probability of having won before n+1 tails is A114604 / A006125 . - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Dec 14 2005

a(n)=A126883(n-1)+1. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2007

M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry. J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97

N. Elkies, G. Kuperberg, M. Larsen and J. Propp, Alternating sign matrices and domino tilings, Journal of Algebraic Combinatorics {\bf 1}, 111-132, 219-234 (1992).

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 178.

D. Grensing, I. Carlsen and H.-Chr. Zapp, Some exact results for the dimer problem on plane lattices with non-standard boundaries, Phil. Mag. A {\bf 41} (1980), 777-781.

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.

F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 178.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 3, Eq. (1.1.2).

W. Jockusch, Perfect matchings and perfect squares. J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.

W. H. Mills, D. P. Robbins and H. Rumsey, Jr., Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A {\bf 34} (1983), 340-359.

J. Propp, Enumeration of matchings: problems and progress, in: New perspectives in geometric combinatorics, L. Billera et al., eds., Mathematical Sciences Research Institute series, vol. 38, Cambridge University Press, 1999.

D. Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939-954.

N. J. A. Sloane, Table of n, a(n) for n = 0..50

F. Ardila and R. P. Stanley, Tilings

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

M. Ciucu, Perfect matchings of cellular graphs, J. Algebraic Combin., 5 (1996) 87-103.

M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97

S.-P. Eu and T.-S. Fu, A simple proof of the Aztec diamond problem

H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects

E. H. Kuo, Applications of graphical condensation for enumerating matchings and tilings

G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics

J. Propp and R. P. Stanley, Domino tilings with barriers

S. S. Skiena, Generating graphs

D. E. Speyer, Perfect matchings and the octahedral recurrence

R. P. Stanley, A combinatorial miscellany

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to dominoes

Sequence is given by the Hankel transform of A001003 (Schroeder's numbers)= 1, 1, 3, 11, 45, 197, 903, ...; example : det([1, 1, 3, 11; 1, 3, 11, 45; 3, 11, 45, 197; 11, 45, 197, 903]) = 2^6 = 64 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 02 2004

a(n)=2^floor(n^2/2)/2^floor(n/2). - Paul Barry (pbarry(AT)wit.ie), Oct 04 2004

seq(2^(binomial(2+n, n)), n=-2..13); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2007

with(finance):seq(mul(futurevalue( 2, 1, k), k=0..n), n=-2..15); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 01 2008

a:=n->mul(2^k, k=0..n): seq(a(n), n=-1..14); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 16 2008

Cf. A000568 for the unlabeled analogue, A006129, A053763, A006253, A004003.

Cf. A001187 (connected labeled graphs).

Cf. A095340, A103904, A005329, A114604.

Sequence in context: A139683 A139684 A139685 this_sequence A006119 A073113 A091794

Adjacent sequences: A006122 A006123 A006124 this_sequence A006126 A006127 A006128

nonn,easy,nice

njas

More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 09 2000