OFFSET
0,5
COMMENTS
As square array: number of certain symmetric plane partitions, see Forrester/Gamburd paper.
Formatted as a square array, the column k gives the Hankel transform of the Catalan numbers (A000108) beginning at A000108(k); example: Hankel transform of [42, 132, 429, 1430, 4862, ...] is [42, 594, 4719, 26026, 111384, ...] (see A091962). - Philippe Deléham, Apr 12 2007
As square array T(n,k): number of all k-watermelons with a wall of length n. - Ralf Stephan, May 09 2007
Consider "Young tableaux with entries from the set {1,...,n}, strictly increasing in rows and not decreasing in columns. Note that usually the reverse convention between rows and columns is used." de Sainte-Catherine and Viennot (1986) proved that "the number b_{n,k} of such Young tableaux having only columns with an even number of elements and bounded by height p = 2*k" is given by b_{n,k} = Product_{1 <= i <= j <= n} (2*k + i + j)/(i + j)." It turns out that for the current array, T(n,k) = b(n-k,k) for n >= 0 and 0 <= k <= n. - Petros Hadjicostas, Sep 04 2019
As square array, b(k, n) = T(n+k-1, n) for k >= 1 and n >= 1 is the number of n-tuples P = (p_1, p_2, ..., p_n) of non-intersecting lattice paths that lie below the diagonal, such that each p_i starts at (i, i) and ends at (2n+k-i, 2n+k-i). (This is just a different way of looking at n-watermelons with a wall of length k since many of the steps of these paths are going to be fixed while the rest form an n-watermelon. See the Krattenthaler et al. paper.) Equivalently b(k, n) is the number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths, each with 2k steps such that for every i (1 <= i <= n-1), p_i is included in p_{i+1}. A Dyck path p is said to be included in a Dyck path q if the height of path p after j steps is at most the height of path q after j steps, for all j (1 <= j <= 2k). - Farzan Byramji, Jun 17 2021
LINKS
Alois P. Heinz, Rows n = 0..100, flattened
R. Bacher, Matrices related to the Pascal triangle, arXiv:math/0109013 [math.CO], 2001.
Paul Barry, Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers, arXiv:2011.10827 [math.CO], 2020.
M. de Sainte-Catherine and G. Viennot, Enumeration of certain Young tableaux with bounded height, in: G. Labelle and P. Leroux (eds), Combinatoire énumérative, Lecture Notes in Mathematics, vol. 1234, Springer, Berlin, Heidelberg, 1986, pp. 58-67.
P. J. Forrester and A. Gamburd, Counting formulas associated with some random matrix averages, arXiv:math/0503002 [math.CO], 2005.
P. J. Forrester and A. Gamburd, Counting formulas associated with some random matrix averages, J. Combin. Theory Ser. A 113(6) (2006), 934-951.
M. Fulmek, Asymptotics of the average height of 2-watermelons with a wall, arXiv:math/0607163 [math.CO], 2006.
M. Fulmek, Asymptotics of the average height of 2-watermelons with a wall, Electron. J. Combin. 14 (2007), R64.
C. Krattenthaler, A. J. Guttmann and X. G. Viennot, Vicious walkers, friendly walkers and Young tableaux: II. With a wall, J. Phys. A: Math. Gen. 33 (2000), 8835-8866.
Vincent Pilaud, Brick polytopes, lattice quotients, and Hopf algebras, arXiv:1505.07665 [math.CO], 2015.
Vincent Pilaud, Brick polytopes, lattice quotients, and Hopf algebras, J. Combin. Theory Ser. A 155 (2018), 418-457.
Michael Somos, Number Walls in Combinatorics.
FORMULA
T(n,k) = Product_{i=1..n-k} Product_{j=i..n-k} (i+j+2*k)/(i+j). [corrected by Petros Hadjicostas, Jul 24 2019]
From G. C. Greubel, Dec 17 2021: (Start)
T(n, k) = Product_{j=0..k-1} binomial(2*n-2*j, n-j)/binomial(n+j+1, n-j).
T(n, k) = ((n+1)!/(n-k+1)!)*Product_{j=0..k-1} Catalan(n-j)/binomial(n+j+1, n-j). (End)
EXAMPLE
Triangle T(n,k) (with rows n >= 0 and columns k >= 0) starts as follows:
1;
1, 1;
1, 2, 1;
1, 5, 3, 1;
1, 14, 14, 4, 1;
1, 42, 84, 30, 5, 1;
1, 132, 594, 330, 55, 6, 1;
1, 429, 4719, 4719, 1001, 91, 7, 1;
1, 1430, 40898, 81796, 26026, 2548, 140, 8, 1;
1, 4862, 379236, 1643356, 884884, 111384, 5712, 204, 9, 1;
...
MAPLE
T:= (n, k)-> mul(mul((i+j+2*k)/(i+j), j=i..n-k), i=1..n-k):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Sep 04 2019
MATHEMATICA
T[n_, k_] := Product[(2*i+1)!*(2*n-2*i)!/(n-i)!/(n+i+1)!, {i, 0, k-1}]; Table[T[n, k], {n, 1, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 28 2015, adapted from PARI *)
PROG
(PARI) T(n, k)=if(k<0 || k>n, 0, prod(i=0, k-1, (2*i+1)!*(2*n-2*i)!/(n-i)!/(n+i+1)!))
(PARI) {C(n)=if(n<0, 0, (2*n)!/n!/(n+1)!)}; T(n, k)=if(k<0 || k>n, 0, matdet(matrix(k, k, i, j, C(i+j-1+n-k))))
(Sage)
def A078920(n, k): return product( binomial(2*n-2*j, n-j)/binomial(n+j+1, n-j) for j in (0..k-1) )
flattened([[A078920(n, k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Dec 17 2021
CROSSREFS
KEYWORD
AUTHOR
Michael Somos, Dec 15 2002
EXTENSIONS
T(0,0) = 1 prepended by Petros Hadjicostas, Jul 24 2019
STATUS
approved