A103905
Square array T(n,k) read by antidiagonals: number of tilings of an hexagon.
1, 1, 2, 1, 6, 3, 1, 20, 20, 4, 1, 70, 175, 50, 5, 1, 252, 1764, 980, 105, 6, 1, 924, 19404, 24696, 4116, 196, 7, 1, 3432, 226512, 731808, 232848, 14112, 336, 8, 1, 12870, 2760615, 24293412, 16818516, 1646568, 41580, 540, 9, 1, 48620, 34763300
Offset: 1
Examples
Array begins: 1, 2, 3, 4, 5, 6, ... 1, 6, 20, 50, 105, 196, ... 1, 20, 175, 980, 4116, 14112, ... 1, 70, 1764, 24696, 232848, 1646568, ... 1, 252, 19404, 731808, 16818516, 267227532, ... ...
Links
- Peter J. Forrester and Alex Gamburd, Counting formulas associated with some random matrix averages, arXiv:math/0503002 [math.CO], 2005.
- Anthony J. Guttmann, Aleksandr L. Owczarek and Xavier G. Viennot, Vicious walkers and Young tableaux. I. Without walls, J. Phys. A 31 (1998) 8123-8135.
- Harald Helfgott and Ira M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
- Christian Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507 [math.CO], 2005.
- Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 28.
- Percy A. MacMahon, Combinatory Analysis, vol. 2, Cambridge University Press, 1916; reprinted by Chelsea, New York, 1960.
Crossrefs
Programs
-
Mathematica
t[n_, k_] := Product[j!*(j + 2*n)!/(j + n)!^2, {j, 0, k - 1}]; Join[{1}, Flatten[ Table[ t[n - k , k], {n, 1, 10}, {k, 1, n}]]] (* Jean-François Alcover, May 16 2012, from 2nd formula *)
Formula
T(n, k) = [V(2n+k-1)V(k-1)V(n-1)^2]/[V(2n-1)V(n+k-1)^2], with V(n) the superfactorial numbers (A000178).
T(n, k) = Prod[j=0..k-1, j!(j+2n)!/(j+n)!^2 ].
T(n, k) = Prod[h=1..n, Prod[i=1..k, Prod[j=1..n, (h+i+j-1)/(h+i+j-2) ]]].
T(n, k) = Prod[i=1..k, Prod[j=n+1..2n+1, i+j]/Prod[j=0..n, i+j]]; - Paul Barry, Jun 13 2006
Conjectural formula as a sum of squares of Vandermonde determinants: T(n,k) = 1/((1!*2! ... *(n-1)!)^2*n!)* sum {1 <= x_1, ..., x_n <= k} (det V(x_1, ..., x_n))^2, where V(x_1, ..., x_n) is the Vandermonde matrix of order n. Compare with A133112. - Peter Bala, Sep 18 2007
For k >= 1, T(n,k)=det(binomial(2*n,n+i-j))1<=i,j<=k [Krattenhaller, Theorem 4].
Let H(n) = product {k = 1..n-1} k!. Then for a,b,c nonnegative integers (H(a)*H(b)*H(c)*H(a+b+c))/(H(a+b)*H(b+c)*H(c+a)) is an integer [MacMahon, Section 4.29 with x -> 1]. Setting a = b = n and c = k gives the entries for this table. - Peter Bala, Dec 22 2011
Comments