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).

1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700, 31095744852375, 12611311859677500, 8639383518297652500, 9995541355448167482000, 19529076234661277104897200, 64427185703425689356896743840, 358869201916137601447486156417296

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• core

Comments

Also known as the Andrews-Mills-Robbins-Rumsey numbers. - N. J. A. Sloane, May 24 2013

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.

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

Examples

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

References

• Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 71, 557, 573.
• D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; A_n on page 4, D_r on page 197.
• C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, Chapter 75, pp. 385-386.
• C. A. Pickover, Wonders of Numbers, "Princeton Numbers", Chapter 83, Oxford Univ. Press NY 2001.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• T. Amdeberhan and V. H. Moll, Arithmetic properties of plane partitions, El. J. Comb. 18 (2) (2011) # P1.
• G. E. Andrews, Plane partitions (III): the Weak Macdonald Conjecture, Invent. Math., 53 (1979), 193-225. (See Theorem 10.)
• Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
• Paul Barry, A Riordan array family for some integrable lattice models, arXiv:2409.09547 [math.CO], 2024.
• Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 7.
• M. T. Batchelor, J. de Gier, and B. Nienhuis, The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions, arXiv:cond-mat/0101385 [cond-mat.stat-mech], 2001.
• Andrew Beveridge, Ian Calaway, and Kristin Heysse, de Finetti Lattices and Magog Triangles, arXiv:1912.12319 [math.CO], 2019.
• E. Beyerstedt, V. H. Moll, and X. Sun, The p-adic Valuation of the ASM Numbers, J. Int. Seq. 14 (2011) # 11.8.7.
• Sara Billey and Matjaž Konvalinka, Generalized rank functions and quilts of alternating sign matrices, arXiv:2412.03236 [math.CO], 2024. See p. 33.
• Sara C. Billey, Brendon Rhoades, and Vasu Tewari, Boolean product polynomials, Schur positivity, and Chern plethysm, arXiv:1902.11165 [math.CO], 2019.
• D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.
• H. Cheballah, S. Giraudo, and R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013-2015.
• M. Ciucu, The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions, J. Combin. Theory Ser. A 86 (1999), 382-389.
• F. Colomo and A. G. Pronko, On the refined 3-enumeration of alternating sign matrices, arXiv:math-ph/0404045, 2004; Advances in Applied Mathematics 34 (2005) 798.
• F. Colomo and A. G. Pronko, Square ice, alternating sign matrices and classical orthogonal polynomials, arXiv:math-ph/0411076, 2004; JSTAT (2005) P01005.
• G. Conant, Magmas and Magog Triangles, 2014.
• J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.
• P. Di Francesco, A refined Razumov-Stroganov conjecture II, arXiv:cond-mat/0409576 [cond-mat.stat-mech], 2004.
• P. Di Francesco, Twenty Vertex model and domino tilings of the Aztec triangle, arXiv:2102.02920 [math.CO], 2021. Mentions this sequence.
• P. Di Francesco, P. Zinn-Justin, and J.-B. Zuber, Determinant formulas for some tiling problems..., arXiv:math-ph/0410002, 2004.
• FindStat - Combinatorial Statistic Finder, Alternating sign matrices
• I. Fischer, The number of monotone triangles with prescribed bottom row, arXiv:math/0501102 [math.CO], 2005.
• Ilse Fischer and Manjil P. Saikia, Refined Enumeration of Symmetry Classes of Alternating Sign Matrices, arXiv:1906.07723 [math.CO], 2019.
• Ilse Fischer and Matjaz Konvalinka, A bijective proof of the ASM theorem, Part I: the operator formula, arXiv:1910.04198 [math.CO], 2019.
• T. Fonseca and F. Balogh, The higher spin generalization of the 6-vertex model with domain wall boundary conditions and Macdonald polynomials, Journal of Algebraic Combinatorics, 2014, arXiv:1210.4527
• D. D. Frey and J. A. Sellers, Jacobsthal Numbers and Alternating Sign Matrices, Journal of Integer Sequences Vol. 3 (2000) #00.2.3.
• D. D. Frey and J. A. Sellers, Prime Power Divisors of the Number of n X n Alternating Sign Matrices
• Markus Fulmek, A statistics-respecting bijection between permutation matrices and descending plane partitions without special parts, Electronic journal of combinatorics, 27(1) (2020), #P1.391.
• M. Gardner, Letter to N. J. A. Sloane, Jun 20 1991.
• C. Heuberger and H. Prodinger, A precise description of the p-adic valuation of the number of alternating sign matrices, Intl. J. Numb. Th. 7 (1) (2011) 57-69.
• 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 arXiv:1611.03387.
• Frederick Huang, The 20 Vertex Model and Related Domino Tilings, Ph. D. Dissertation, UC Berkeley, 2023. See p. 1.
• Hassan Isanloo, The volume and Ehrhart polynomial of the alternating sign matrix polytope, Cardiff University (Wales, UK 2019).
• Masato Kobayashi, Weighted counting of inversions on alternating sign matrices, arXiv:1904.02265 [math.CO], 2019.
• G. Kuperberg, Another proof of the alternating-sign matrix conjecture, arXiv:math/9712207 [math.CO], 1997; Internat. Math. Res. Notices, No. 3, (1996), 139-150.
• G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001; Ann. Math. 156 (3) (2002) 835-866
• W. H. Mills, David P Robbins, and Howard Rumsey Jr., Alternating sign matrices and descending plane partitions J. Combin. Theory Ser. A 34 (1983), no. 3, 340--359. MR0700040 (85b:05013).
• Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
• C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, Chapter 75, pp. 385-386. [Annotated scanned copy]
• J. Propp, The many faces of alternating-sign matrices, Discrete Mathematics and Theoretical Computer Science Proceedings AA (DM-CCG), 2001, 43-58.
• A. V. Razumov and Yu. G. Stroganov, Spin chains and combinatorics, arXiv:cond-mat/0012141 [cond-mat.stat-mech], 2000.
• Lukas Riegler, Simple enumeration formulas related to Alternating Sign Monotone Triangles and standard Young tableaux, Dissertation, Universitat Wien, 2014.
• D. P. Robbins, The story of 1, 2, 7, 42, 429, 7436, ..., Math. Intellig., 13 (No. 2, 1991), 12-19.
• D. P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math/0008045 [math.CO], 2000.
• R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
• R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]
• Yu. G. Stroganov, 3-enumerated alternating sign matrices, arXiv:math-ph/0304004, 2003.
• X. Sun and V. H. Moll, The p-adic Valuations of Sequences Counting Alternating Sign Matrices, JIS 12 (2009) 09.3.8.
• Eric Weisstein's World of Mathematics, Alternating Sign Matrix
• Eric Weisstein's World of Mathematics, Descending Plane Partition
• D. Zeilberger, Proof of the alternating-sign matrix conjecture, arXiv:math/9407211 [math.CO], 1994.
• D. Zeilberger, Proof of the alternating-sign matrix conjecture, Elec. J. Combin., Vol. 3 (Number 2) (1996), #R13.
• D. Zeilberger, Proof of the Refined Alternating Sign Matrix Conjecture, arXiv:math/9606224 [math.CO], 1996.
• D. Zeilberger, A constant term identity featuring the ubiquitous (and mysterious) Andrews-Mills-Robbins-Ramsey numbers 1,2,7,42,429,..., J. Combin. Theory, A 66 (1994), 17-27. The link is to a comment on Doron Zeilberger's home page. A backup copy is here [pdf file only, no active links]
• D. Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939-954. The link is to a version on Doron Zeilberger's home page. A backup copy is here [pdf file only, no active links]
• Paul Zinn-Justin, Integrability and combinatorics, arXiv:2404.13221 [math.CO], 2024. See p. 12.
• Index entries for sequences related to factorial numbers
• Index entries for "core" sequences

Crossrefs

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

Programs

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

Maple

A005130 := proc(n) local k; mul((3*k+1)!/(n+k)!,k=0..n-1); end;
# Bill Gosper's approximation (for n>0):
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

Mathematica

f[n_] := Product[(3k + 1)!/(n + k)!, {k, 0, n - 1}]; Table[ f[n], {n, 0, 17}] (* Robert G. Wilson v, Jul 15 2004 *)
a[ n_] := If[ n < 0, 0, Product[(3 k + 1)! / (n + k)!, {k, 0, n - 1}]]; (* Michael Somos, May 06 2015 *)

PARI

{a(n) = if( n<0, 0, prod(k=0, n-1, (3*k + 1)! / (n + k)!))}; /* Michael Somos, Aug 30 2003 */

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 */

Python

from math import prod, factorial
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

Formula

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

The Hankel transform of A025748 is a(n) * 3^binomial(n, 2). - Michael Somos, Aug 30 2003

a(n) = sqrt(A049503).

From Bill Gosper, Mar 11 2014: (Start)

A "Stirling's formula" for this sequence is

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).

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

1.0063254118710128, 2.003523267231662,

7.0056223910285915, 42.01915917750558,

429.12582410098327, 7437.518404899576,

218380.8077275304, 1.085146545456063*^7,

9.119184824937415*^8

(End)

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

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