A005130 - OEIS

%I M1808 #225 Dec 10 2024 12:32:53

%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 a(n) is odd iff n is a Jacobsthal number (A001045) [Frey and Sellers, 2000]. - _Gary W. Adamson_, May 27 2009

%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 71, 557, 573.

%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 and V. H. Moll, <a href="https://doi.org/10.37236/1997">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 Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.

%H Paul Barry, <a href="https://arxiv.org/abs/2409.09547">A Riordan array family for some integrable lattice models</a>, arXiv:2409.09547 [math.CO], 2024.

%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 Andrew Beveridge, Ian Calaway, and Kristin Heysse, <a href="https://arxiv.org/abs/1912.12319">de Finetti Lattices and Magog Triangles</a>, arXiv:1912.12319 [math.CO], 2019.

%H E. Beyerstedt, V. H. Moll, and 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 Sara Billey and Matjaž Konvalinka, <a href="https://arxiv.org/abs/2412.03236">Generalized rank functions and quilts of alternating sign matrices</a>, arXiv:2412.03236 [math.CO], 2024. See p. 33.

%H Sara C. Billey, Brendon Rhoades, and Vasu Tewari, <a href="https://arxiv.org/abs/1902.11165">Boolean product polynomials, Schur positivity, and Chern plethysm</a>, arXiv:1902.11165 [math.CO], 2019.

%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, and 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>, arXiv:math-ph/0404045, 2004; 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>, arXiv:math-ph/0411076, 2004; JSTAT (2005) P01005.

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

%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 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, <a href="https://arxiv.org/abs/2102.02920">Twenty Vertex model and domino tilings of the Aztec triangle</a>, arXiv:2102.02920 [math.CO], 2021. Mentions this sequence.

%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 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 Ilse Fischer and Manjil P. Saikia, <a href="https://arxiv.org/abs/1906.07723">Refined Enumeration of Symmetry Classes of Alternating Sign Matrices</a>, arXiv:1906.07723 [math.CO], 2019.

%H Ilse Fischer and Matjaz Konvalinka, <a href="https://arxiv.org/abs/1910.04198">A bijective proof of the ASM theorem, Part I: the operator formula</a>, arXiv:1910.04198 [math.CO], 2019.

%H T. Fonseca and 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 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 Markus Fulmek, <a href="https://doi.org/10.37236/8265">A statistics-respecting bijection between permutation matrices and descending plane partitions without special parts</a>, Electronic journal of combinatorics, 27(1) (2020), #P1.391.

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

%H C. Heuberger and 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 Dylan Heuer, 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 Frederick Huang, <a href="https://escholarship.org/uc/item/7p96n76z">The 20 Vertex Model and Related Domino Tilings</a>, Ph. D. Dissertation, UC Berkeley, 2023. See p. 1.

%H Hassan Isanloo, <a href="https://orca.cardiff.ac.uk/id/eprint/125150">The volume and Ehrhart polynomial of the alternating sign matrix polytope</a>, Cardiff University (Wales, UK 2019).

%H Masato Kobayashi, <a href="https://arxiv.org/abs/1904.02265">Weighted counting of inversions on alternating sign matrices</a>, arXiv:1904.02265 [math.CO], 2019.

%H G. Kuperberg, <a href="https://arxiv.org/abs/math/9712207">Another proof of the alternating-sign matrix conjecture</a>, arXiv:math/9712207 [math.CO], 1997; 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/0008184 [math.CO], 2000-2001; <a href="http://dx.doi.org/10.2307/3597283">Ann. Math. 156 (3) (2002) 835-866</a>

%H W. H. Mills, David P Robbins, and 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>, Discrete Mathematics and Theoretical Computer Science Proceedings AA (DM-CCG), 2001, 43-58.

%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 [cond-mat.stat-mech], 2000.

%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>, arXiv:math-ph/0304004, 2003.

%H X. Sun and 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="https://doi.org/10.37236/1271">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. The link is to a comment on Doron Zeilberger's home page. A backup copy is <a href="/A005130/a005130_3.pdf">here</a> [pdf file only, no active links]

%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. The link is to a version on Doron Zeilberger's home page. A backup copy is <a href="/A005130/a005130_4.pdf">here</a> [pdf file only, no active links]

%H Paul Zinn-Justin, <a href="https://arxiv.org/abs/2404.13221">Integrability and combinatorics</a>, arXiv:2404.13221 [math.CO], 2024. See p. 12.

%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

%F For n>0, a(n) = 3^(n - 1/3) * BarnesG(n+1) * BarnesG(3*n)^(1/3) * Gamma(n)^(1/3) * Gamma(n + 1/3)^(2/3) / (BarnesG(2*n+1) * Gamma(1/3)^(2/3)). - _Vaclav Kotesovec_, Mar 04 2021

%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 */

%o (GAP) a:=List([0..18],n->Product([0..n-1],k->Factorial(3*k+1)/Factorial(n+k)));; Print(a); # _Muniru A Asiru_, Jan 02 2019

%o (Python)

%o from math import prod, factorial

%o def A005130(n): return prod(factorial(3*k+1) for k in range(n))//prod(factorial(n+k) for k in range(n)) # _Chai Wah Wu_, Feb 02 2022

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

%K nonn,easy,nice,core,changed

%O 0,3

%A _N. J. A. Sloane_