A187783 De Bruijn's triangle, T(m,n) = (m*n)!/(n!^m) read by downward antidiagonals.
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 6, 1, 1, 1, 20, 90, 24, 1, 1, 1, 70, 1680, 2520, 120, 1, 1, 1, 252, 34650, 369600, 113400, 720, 1, 1, 1, 924, 756756, 63063000, 168168000, 7484400, 5040, 1
Offset: 0
Examples
T(3,5) = (3*5)!/(5!^3) = 756756 = A014609(3) = A006480(5) is the number of permutations of a multiset that contains 3 different elements 5 times, e.g., {1,1,1,1,1,2,2,2,2,2,3,3,3,3,3}.
Links
- Tilman Piesk, First 54 rows of the triangle, flattened
- T. Chappell, A. Lascoux, S. Ole Warnaar, and W. Zudilin, Logarithmic and complex constant term identities, arXiv:1112.3130 [math.CO], 2012.
- Tilman Piesk, Array for indices 0..16
- Tilman Piesk, PHP code used to create the b-file
- Tilman Piesk, Illustration of the multisets for m,n=0..4
- Wikipedia, Permutations of multisets, Pascal matrix and simplex
Crossrefs
Programs
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Magma
[Factorial(k*(n-k))/(Factorial(n-k))^k: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 26 2022
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Mathematica
T[n_, k_]:= (k*n)!/(n!)^k; Table[T[n, k-n], {k, 9}, {n, 0, k-1}]//Flatten -
SageMath
def A187783(n,k): return gamma(k*(n-k)+1)/(factorial(n-k))^k flatten([[A187783(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Dec 26 2022
Formula
T(m,n) = (m*n)!/(n!)^m.
A060540(m,n) = T(m,n)/m! . - R. J. Mathar, Jun 21 2023
Extensions
Row m=0 prepended by Tilman Piesk, Oct 28 2014
Comments