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
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
KEYWORD
nonn,easy,walk,nice
AUTHOR
EXTENSIONS
Added a(0), merged A151334 into this one. - N. J. A. Sloane, Feb 24 2016
STATUS
approved