Search: id:A006125 Results 1-1 of 1 results found. %I A006125 M1897 %S A006125 1,1,2,8,64,1024,32768,2097152,268435456,68719476736,35184372088832, %T A006125 36028797018963968,73786976294838206464,302231454903657293676544, %U A006125 2475880078570760549798248448,40564819207303340847894502572032 %N A006125 2^{n(n-1)/2}. %C A006125 Number of graphs on n labeled nodes; also number of outcomes of labeled n-team round-robin tournaments. %C A006125 Number of perfect matchings of order n Aztec diamond [see Speyer] %C A006125 Number of Gelfand-Zeitlin patterns with bottom row [1,2,3,...,n]. [Zeilberger] %C A006125 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 %C A006125 Comment from Jim Propp (propp(AT)math.wisc.edu): a(n) is the number of ways to tile the region %C A006125 .........o-----o %C A006125 .........|.....| %C A006125 ......o--o.....o--o %C A006125 ......|...........| %C A006125 ...o--o...........o--o %C A006125 ...|.................| %C A006125 o--o.................o--o %C A006125 |.......................| %C A006125 |.......................| %C A006125 |.......................| %C A006125 o--o.................o--o %C A006125 ...|.................| %C A006125 ...o--o...........o--o %C A006125 ......|...........| %C A006125 ......o--o.....o--o %C A006125 .........|.....| %C A006125 .........o-----o %C A006125 (top-to-bottom distance = 2n) with dominoes like either of %C A006125 o--o o-----o %C A006125 |..| |.....| %C A006125 |..| o-----o %C A006125 |..| %C A006125 o--o %C A006125 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 %C A006125 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 %C A006125 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 %C A006125 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 %C A006125 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 %C A006125 a(n)=A126883(n-1)+1. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2007 %C A006125 Equals right border of triangle A158474(unsigned). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 20 2009] %D A006125 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A006125 M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry. J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97 %D A006125 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). %D A006125 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 178. %D A006125 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. %D A006125 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517. %D A006125 F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 178. %D A006125 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 3, Eq. (1.1.2). %D A006125 W. Jockusch, Perfect matchings and perfect squares. J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115. %D A006125 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. %D A006125 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 A006125 D. Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939-954. %H A006125 N. J. A. Sloane, Table of n, a(n) for n = 0..50 %H A006125 F. Ardila and R. P. Stanley, Tilings %H A006125 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A006125 M. Ciucu, Perfect matchings of cellular graphs, J. Algebraic Combin., 5 (1996) 87-103. %H A006125 M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97 %H A006125 S.-P. Eu and T.-S. Fu, A simple proof of the Aztec diamond problem %H A006125 H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects %H A006125 E. H. Kuo, Applications of graphical condensation for enumerating matchings and tilings %H A006125 G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2. %H A006125 J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics %H A006125 J. Propp and R. P. Stanley, Domino tilings with barriers %H A006125 S. S. Skiena, Generating graphs %H A006125 D. E. Speyer, Perfect matchings and the octahedral recurrence %H A006125 R. P. Stanley, A combinatorial miscellany %H A006125 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A006125 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A006125 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A006125 Index entries for sequences related to dominoes %F A006125 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 %F A006125 a(n)=2^floor(n^2/2)/2^floor(n/2). - Paul Barry (pbarry(AT)wit.ie), Oct 04 2004 %p A006125 seq(2^(binomial(2+n,n)), n=-2..13); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2007 %p A006125 with(finance):seq(mul(futurevalue( 2, 1, k),k=0..n),n=-2..15); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 01 2008 %p A006125 a:=n->mul(2^k, k=0..n): seq(a(n), n=-1..14); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 16 2008 %p A006125 restart:a:= proc(n) option remember; if n=0 then 1 else add (binomial (n-1,j) *a(n-1), j=0..n-1) fi end: seq (a(n), n=0..15);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2009] %Y A006125 Cf. A000568 for the unlabeled analogue, A006129, A053763, A006253, A004003. %Y A006125 Cf. A001187 (connected labeled graphs). %Y A006125 Cf. A095340, A103904, A005329, A114604. %Y A006125 A158474 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 20 2009] %Y A006125 Sequence in context: A139683 A139684 A139685 this_sequence A006119 A073113 A091794 %Y A006125 Adjacent sequences: A006122 A006123 A006124 this_sequence A006126 A006127 A006128 %K A006125 nonn,easy,nice %O A006125 0,3 %A A006125 N. J. A. Sloane (njas(AT)research.att.com). %E A006125 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 09 2000 Search completed in 0.002 seconds