login
This site is supported by donations to The OEIS Foundation.

 

Logo

The October issue of the Notices of the Amer. Math. Soc. has an article about the OEIS.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005130 Robbins numbers: a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!; also the number of descending plane partitions whose parts do not exceed n; also the number of n X n alternating sign matrices (ASM's).
(Formerly M1808)
45

%I M1808

%S 1,1,2,7,42,429,7436,218348,10850216,911835460,129534272700,

%T 31095744852375,12611311859677500,8639383518297652500,

%U 9995541355448167482000,19529076234661277104897200,64427185703425689356896743840,358869201916137601447486156417296

%N Robbins numbers: a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!; also the number of descending plane partitions whose parts do not exceed n; also the number of n X n alternating sign matrices (ASM's).

%C Also known as the Andrews-Mills-Robbins-Rumsey numbers. - _N. J. A. Sloane_, May 24 2013

%C An alternating sign matrix is a matrix of 0's, 1's and -1's such that (a) the sum of each row and column is 1; (b) the nonzero entries in each row and column alternate in sign.

%C From _Gary W. Adamson_, May 27 2009: (Start)

%C Starting with offset 1 = row sums of triangle A160708, and convolution square of A160707.

%C a(n) is odd iff n is a Jacobsthal number [Frey and Sellers, 2000].

%C Starting with offset 1 = row sums of triangle A160708.

%C Starting (1, 2, 7,...) = convolution square of A160707: [1, 1, 3, 18, 192,...].

%C (End)

%D D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; A_n on page 4, D_r on page 197.

%D C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, Chapter 75, pp. 385-386.

%D C. A. Pickover, Wonders of Numbers, "Princeton Numbers", Chapter 83, Oxford Univ. Press NY 2001.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A005130/b005130.txt">Table of n, a(n) for n = 0..100</a>

%H T. Amdeberhan, V. H. Moll, <a href="http://www.emis.ams.org/journals/EJC/Volume_18/Abstracts/v18i2p1.html">Arithmetic properties of plane partitions</a>, El. J. Comb. 18 (2) (2011) # P1.

%H G. E. Andrews, <a href="http://dx.doi.org/10.1007/BF01389763">Plane partitions (III): the Weak Macdonald Conjecture</a>, Invent. Math., 53 (1979), 193-225. (See Theorem 10.)

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.

%H M. T. Batchelor, J. de Gier and B. Nienhuis, <a href="https://arxiv.org/abs/cond-mat/0101385">The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions</a>, arXiv:cond-mat/0101385 [cond-mat.stat-mech], 2001.

%H E. Beyerstedt, V. H. Moll, X. Sun, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Moll/moll2.html">The p-adic Valuation of the ASM Numbers</a>, J. Int. Seq. 14 (2011) # 11.8.7

%H D. M. Bressoud and J. Propp, <a href="http://www.ams.org/notices/199906/fea-bressoud.pdf">How the alternating sign matrix conjecture was solved</a>, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.

%H H. Cheballah, S. Giraudo, R. Maurice, <a href="https://arxiv.org/abs/1306.6605">Combinatorial Hopf algebra structure on packed square matrices</a>, arXiv preprint arXiv:1306.6605 [math.CO], 2013-2015.

%H M. Ciucu, <a href="http://dx.doi.org/10.1006/jcta.1998.2922">The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions</a>, J. Combin. Theory Ser. A 86 (1999), 382-389.

%H F. Colomo and A. G. Pronko, <a href="http://arXiv.org/abs/math-ph/0404045">On the refined 3-enumeration of alternating sign matrices</a>, Advances in Applied Mathematics 34 (2005) 798.

%H F. Colomo and A. G. Pronko, <a href="https://arxiv.org/abs/math-ph/0411076">Square ice, alternating sign matrices and classical orthogonal polynomials</a>, JSTAT (2005) P01005.

%H G. Conant, <a href="http://homepages.math.uic.edu/~gconant/Math/magmas.pdf">Magmas and Magog Triangles</a>, 2014.

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/AlternatingSignMatrices">Alternating sign matrices</a>

%H I. Fischer, <a href="https://arxiv.org/abs/math/0501102">The number of monotone triangles with prescribed bottom row</a>, arXiv:math/0501102 [math.CO], 2005.

%H T. Fonseca, F. Balogh, <a href="http://dx.doi.org/10.1007/s10801-014-0555-0">The higher spin generalization of the 6-vertex model with domain wall boundary conditions and Macdonald polynomials</a>, Journal of Algebraic Combinatorics, 2014, <a href="http://arxiv.org/abs/1210.4527">arXiv:1210.4527</a>

%H P. Di Francesco, <a href="https://arxiv.org/abs/cond-mat/0409576">A refined Razumov-Stroganov conjecture II</a>, arXiv:cond-mat/0409576 [cond-mat.stat-mech], 2004.

%H P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, <a href="https://arxiv.org/abs/math-ph/0410002">Determinant formulas for some tiling problems...</a>, arXiv:math-ph/0410002, 2004.

%H D. D. Frey and J. A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SELLERS/sellers.pdf">Jacobsthal Numbers and Alternating Sign Matrices</a>, Journal of Integer Sequences Vol. 3 (2000) #00.2.3.

%H D. D. Frey and J. A. Sellers, <a href="http://www.math.psu.edu/sellersj/p23.pdf">Prime Power Divisors of the Number of n X n Alternating Sign Matrices</a>

%H M. Gardner, <a href="/A005130/a005130_1.pdf">Letter to N. J. A. Sloane</a>, Jun 20 1991.

%H J. de Gier, <a href="https://arxiv.org/abs/math/0211285">Loops, matchings and alternating-sign matrices</a>, arXiv:math/0211285 [math.CO], 2002-2003.

%H C. Heuberger, H. Prodinger, <a href="http://dx.doi.org/10.1142/S1793042111003892">A precise description of the p-adic valuation of the number of alternating sign matrices</a>, Intl. J. Numb. Th. 7 (1) (2011) 57-69

%H Heuer, Dylan, Chelsey Morrow, Ben Noteboom, Sara Solhjem, Jessica Striker, and Corey Vorland. "Chained permutations and alternating sign matrices - Inspired by three-person chess." Discrete Mathematics 340, no. 12 (2017): 2732-2752. Also <a href="http://arxiv.org/abs/1611.03387">arXiv:1611.03387</a>.

%H G. Kuperberg, <a href="http://arXiv.org/abs/math.CO/9712207">Another proof of the alternating-sign matrix conjecture</a>, Internat. Math. Res. Notices, No. 3, (1996), 139-150.

%H G. Kuperberg, <a href="https://arxiv.org/abs/math/0008184">Symmetry classes of alternating-sign matrices under one roof</a>, arXiv math.CO/0008184, <a href="http://dx.doi.org/10.2307/3597283">Ann. Math. 156 (3) (2002) 835-866</a>

%H W. H. Mills, David P Robbins, Howard Rumsey Jr., <a href="http://dx.doi.org/10.1016/0097-3165(83)90068-7">Alternating sign matrices and descending plane partitions</a> J. Combin. Theory Ser. A 34 (1983), no. 3, 340--359. MR0700040 (85b:05013)

%H Igor Pak, <a href="https://arxiv.org/abs/1803.06636">Complexity problems in enumerative combinatorics</a>, arXiv:1803.06636 [math.CO], 2018.

%H C. Pickover, <a href="/A005130/a005130_2.pdf">Mazes for the Mind</a>, St. Martin's Press, NY, 1992, Chapter 75, pp. 385-386. [Annotated scanned copy]

%H J. Propp, <a href="http://www.dmtcs.org/pdfpapers/dmAA0103.pdf">The many faces of alternating-sign matrices.</a>

%H A. V. Razumov and Yu. G. Stroganov, <a href="http://arXiv.org/abs/cond-mat/0012141">Spin chains and combinatorics</a>, arXiv cond-mat/0012141

%H Lukas Riegler, <a href="http://homepage.univie.ac.at/lukas.riegler/docs/Riegler-Dissertation.pdf">Simple enumeration formulas related to Alternating Sign Monotone Triangles and standard Young tableaux</a>, Dissertation, Universitat Wien, 2014.

%H D. P. Robbins, <a href="http://dx.doi.org/10.1007/BF03024081">The story of 1, 2, 7, 42, 429, 7436, ...</a>, Math. Intellig., 13 (No. 2, 1991), 12-19.

%H D. P. Robbins, <a href="https://arxiv.org/abs/math/0008045">Symmetry classes of alternating sign matrices</a>, arXiv:math/0008045 [math.CO], 2000.

%H R. P. Stanley, <a href="http://dx.doi.org/10.1007/BFb0072521">A baker's dozen of conjectures concerning plane partitions</a>, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.

%H R. P. Stanley, <a href="/A005130/a005130.pdf">A baker's dozen of conjectures concerning plane partitions</a>, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]

%H Yu. G. Stroganov, <a href="http://arXiv.org/abs/math-ph/0304004">3-enumerated alternating sign matrices</a>

%H X. Sun, V. H. Moll, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Moll/moll.html">The p-adic Valuations of Sequences Counting Alternating Sign Matrices</a>, JIS 12 (2009) 09.3.8

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlternatingSignMatrix.html">Alternating Sign Matrix</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DescendingPlanePartition.html">Descending Plane Partition</a>

%H D. Zeilberger, <a href="https://arxiv.org/abs/math/9407211">Proof of the alternating-sign matrix conjecture</a>, arXiv:math/9407211 [math.CO], 1994.

%H D. Zeilberger, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i2r13">Proof of the alternating-sign matrix conjecture</a>, Elec. J. Combin., Vol. 3 (Number 2) (1996), #R13.

%H D. Zeilberger, <a href="https://arxiv.org/abs/math/9606224">Proof of the Refined Alternating Sign Matrix Conjecture</a>, arXiv:math/9606224 [math.CO], 1996.

%H D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/amrr.html">A constant term identity featuring the ubiquitous(and mysterious)Andrews-Mills-Robbins-Ramsey numbers 1,2,7,42,429,...</a>, J. Combin. Theory, A 66 (1994), 17-27.

%H D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/DaveRobbins/guess.html">Dave Robbins's Art of Guessing</a>, Adv. in Appl. Math. 34 (2005), 939-954.

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!.

%F The Hankel transform of A025748 is a(n) * 3^binomial(n, 2). - _Michael Somos_, Aug 30 2003

%F a(n) = sqrt(A049503).

%F From _Bill Gosper_, Mar 11 2014: (Start)

%F A "Stirling's formula" for this sequence is

%F a(n) ~ 3^(5/36+(3/2)*n^2)/(2^(1/4+2*n^2)*n^(5/36))*(exp(zeta'(-1))*gamma(2/3)^2/Pi)^(1/3).

%F which gives results which are very close to the true values:

%F 1.0063254118710128, 2.003523267231662,

%F 7.0056223910285915, 42.01915917750558,

%F 429.12582410098327, 7437.518404899576,

%F 218380.8077275304, 1.085146545456063*^7,

%F 9.119184824937415*^8

%F (End)

%F a(n+1) = a(n) * n! * (3*n+1)! / ((2*n)! * (2*n+1)!). - _Reinhard Zumkeller_, Sep 30 2014; corrected by _Eric W. Weisstein_, Nov 08 2016

%e G.f. = 1 + x + 2*x^2 + 7*x^3 + 42*x^4 + 429*x^5 + 7436*x^6 + 218348*x^7 + ...

%p A005130 := proc(n) local k; mul((3*k+1)!/(n+k)!,k=0..n-1); end;

%p # _Bill Gosper_'s approximation (for n>0):

%p a_prox := n -> (2^(5/12-2*n^2)*3^(-7/36+1/2*(3*n^2))*exp(1/3*Zeta(1,-1))*Pi^(1/3)) /(n^(5/36)*GAMMA(1/3)^(2/3)); # _Peter Luschny_, Aug 14 2014

%t f[n_] := Product[(3k + 1)!/(n + k)!, {k, 0, n - 1}]; Table[ f[n], {n, 0, 17}] (* _Robert G. Wilson v_, Jul 15 2004 *)

%t a[ n_] := If[ n < 0, 0, Product[(3 k + 1)! / (n + k)!, {k, 0, n - 1}]]; (* _Michael Somos_, May 06 2015 *)

%o (PARI) {a(n) = if( n<0, 0, prod(k=0, n-1, (3*k + 1)! / (n + k)!))}; /* _Michael Somos_, Aug 30 2003 */

%o (PARI) {a(n) = my(A); if( n<0, 0, A = Vec( (1 - (1 - 9*x + O(x^(2*n)))^(1/3)) / (3*x)); matdet( matrix(n, n, i, j, A[i+j-1])) / 3^binomial(n,2))}; /* _Michael Somos_, Aug 30 2003 */

%Y Cf. A006366, A048601, also A003827, A005156, A005158, A005160-A005164, A050204, A049503, A160707, A160708, A194827, A227833.

%K nonn,easy,nice,core

%O 0,3

%A _N. J. A. Sloane_

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 23 06:51 EDT 2018. Contains 315273 sequences. (Running on oeis4.)