A005789 3-dimensional Catalan numbers. (Formerly M3997)

1, 1, 5, 42, 462, 6006, 87516, 1385670, 23371634, 414315330, 7646001090, 145862174640, 2861142656400, 57468093927120, 1178095925505960, 24584089974896430, 521086299271824330, 11198784501894470250, 243661974372798631650, 5360563436201569896300

Offset

0,3

Comments

Number of standard tableaux of shape (n,n,n). - Emeric Deutsch, May 13 2004

Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of 3 n steps taken from {(-1, 0), (0, 1), (1, -1)}. - Manuel Kauers, Nov 18 2008

Number of up-down permutations of length 2n with no four-term increasing subsequence, or equivalently the number of down-up permutations of length 2n with no four-term decreasing subsequence. (An up-down permutation is one whose descent set is {2, 4, 6, ...}.) - Joel B. Lewis, Oct 04 2009

Equivalent to the number of standard tableaux: number of rectangular arrangements of [1..3n] into n increasing sequences of size 3 and 3 increasing sequences of size n. a(n) counts a subset of A025035(n). - Olivier Gérard, Feb 15 2011

Number of walks in 3-dimensions using steps (1,0,0), (0,1,0), and (0,0,1) from (0,0,0) to (n,n,n) such that after each step we have z>=y>=x. - Thotsaporn Thanatipanonda, Feb 21 2012

Number of words consisting of n 'x' letters, n 'y' letters and n 'z' letters such that the 'x' count is always greater than or equal to the 'y' count and the 'y' count is always greater than or equal to the 'z' count; e.g., for n=2 we have xxyyzz, xxyzyz, xyxyzz, xyxzyz and xyzxyz. - Jon Perry, Nov 16 2012 [here "count" is meant as "number of symbols in any prefix", Joerg Arndt, Jan 02 2024]

References

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

Snover, Stephen L.; and Troyer, Stephanie F.; A four-dimensional Catalan formula. Proceedings of the Nineteenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1989). Congr. Numer. 75 (1990), 123-126.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..700

Joerg Arndt, The a(3)=42 Young tableaux of shape [3,3,3].

Nicolas Borie, Three-dimensional Catalan numbers and product-coproduct prographs, arXiv:1704.00212 [math.CO], 2017.

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.

Paul Drube, Maxwell Krueger, Ashley Skalsky, and Meghan Wren, Set-Valued Young Tableaux and Product-Coproduct Prographs, arXiv:1710.02709 [math.CO], 2017.

Shalosh B. Ekhad and Doron Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux. Also arXiv preprint arXiv:1202.6229, 2012. - N. J. A. Sloane, Jul 07 2012

K. Gorska and K. A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008 [math.CO], 2013.

Martin Griffiths and Nick Lord, The hook-length formula and generalised Catalan numbers, The Mathematical Gazette Vol. 95, No. 532 (March 2011), pp. 23-30

R. Kenyon, J. Miller, S. Sheffield, and D. B. Wilson, Bipolar orientations on planar maps and SLE_12, arXiv preprint arXiv:1511.04068 [math.PR], 2015. Also The Annals of Probability (2019) Vol. 47, No. 3, 1240-1269.

J. B. Lewis, Pattern avoidance and RSK-like algorithms for alternating permutations and Young tableaux, arXiv:0909.4966 [math.CO], 2009-2011. [From Joel B. Lewis, Oct 04 2009]

J. B. Lewis, Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux, Ph. D. Dissertation, Department of Mathematics, MIT, 2012. - From N. J. A. Sloane, Oct 12 2012

Andrew Lohr, Several Topics in Experimental Mathematics, arXiv:1805.00076 [math.CO], 2018.

Michaël Moortgat, The Tamari order for D^3 and derivability in semi-associative Lambek-Grishin Calculus, 15th Workshop: Computational Logic and Applications (CLA 2020).

Maya Sankar, Further Bijections to Pattern-Avoiding Valid Hook Configurations, arXiv:1910.08895 [math.CO], 2019.

S. Snover, Letter to N. J. A. Sloane, May 1991

R. A. Sulanke, Three-dimensional Narayana and Schröder numbers, Theoretical Computer Science, Volume 346, Issues 2-3, 28 November 2005, Pages 455-468.

R. A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54, 20 pp. (see page 16)

S. F. Troyer and S. L. Snover, m-Dimensional Catalan numbers, Preprint, 1989. (Annotated scanned copy)

Wolfgang Unger, Combinatorics of Lattice QCD at Strong Coupling, arXiv:1411.4493 [hep-lat], 2014.

Manuel Wettstein, Trapezoidal Diagrams, Upward Triangulations, and Prime Catalan Numbers, arXiv:1602.07235 [cs.CG], 2016 and Discr. Comp. Geom. 58 (2017) 505-525.

Sherry H. F. Yan, On Wilf equivalence for alternating permutations, Elect. J. Combinat.; 20 (2013), #P58.

Formula

a(n) = 2*(3*n)!/(n!*(n+1)!*(n+2)!).

a(n) = 0!*1!*..*(k-1)! *(k*n)! / ( n!*(n+1)!*..*(n+k-1)! ) for k=3.

G.f.: (1/30)*(1/x-27)*(9*hypergeom([1/3, 2/3],[1],27*x)+(216*x+1)* hypergeom([4/3, 5/3],[2],27*x))-1/(3*x). - Mark van Hoeij, Oct 14 2009

a(n) ~ 3^(3*n+1/2) / (Pi*n^4). - Vaclav Kotesovec, Nov 13 2014

a(n) = 2*A001700(n+1)*A001764(n+1)/(3*(3*n+1)*(3*n+2)). - R. J. Mathar, Aug 10 2015

D-finite with recurrence (n+2)*(n+1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Aug 10 2015

G.f.: x*3F2(4/3,5/3,1;4,3;27x). - R. J. Mathar, Aug 10 2015

E.g.f.: 2F2(1/3,2/3; 2,3; 27*x). - Ilya Gutkovskiy, Oct 13 2017

Maple

a:= n-> (3*n)! *mul(i!/(n+i)!, i=0..2):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 23 2012

Mathematica

Needs["Combinatorica`"]
Table[ NumberOfTableaux@ {n, n, n}], {n, 17}] (* Robert G. Wilson v, Nov 15 2006 *)
Table[2*(3*n)!/(n!*(n+1)!*(n+2)!), {n, 1, 20}] (* Vaclav Kotesovec, Nov 13 2014 *)
aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[aux[0, 0, 3 n], {n, 0, 25}] (* Manuel Kauers, Nov 18 2008 *)

Prog

(Magma) [2*Factorial(3*n)/(Factorial(n)*Factorial(n+1)*Factorial(n+2)): n in [0..20]]; // Vincenzo Librandi, Oct 14 2017
(PARI) a(n) = 2*(3*n)!/(n!*(n+1)!*(n+2)!); \\ Altug Alkan, Mar 14 2018

Crossrefs

A row of A060854.

A subset of A025035.

See A268538 for primitive terms.

Sequence in context: A217808 A217810 A151334 * A217809 A360578 A317352

Adjacent sequences: A005786 A005787 A005788 * A005790 A005791 A005792

Keyword

nonn,easy,walk,nice

Author

N. J. A. Sloane

Extensions

Added a(0), merged A151334 into this one. - N. J. A. Sloane, Feb 24 2016

Status

approved