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

Author

Robert G. Wilson v, Jan 05 2013

Keywords

• nonn
• tabl
• easy

Comments

From Tilman Piesk, Oct 28 2014: (Start)

Number of permutations of a multiset that contains m different elements n times. These multisets have the signatures A249543(m,n-1) for m>=1 and n>=2.

In an m-dimensional Pascal tensor (the generalization of a symmetric Pascal matrix) P(x1,...,xn) = (x1+...+xn)!/(x1!*...*xn!), so the main diagonal of an m-dimensional Pascal tensor is D(n) = (m*n)!/(n!^m). These diagonals are the rows of this array (with m>0), which begins like this:

m\n:0 1 2 3 4 5

0: 1 1 1 1 1 1 ... A000012;

1: 1 1 1 1 1 1 ... A000012;

2: 1 2 6 20 70 252 ... A000984;

3: 1 6 90 1680 34650 756756 ... A006480;

4: 1 24 2520 369600 63063000 11732745024 ... A008977;

5: 1 120 113400 168168000 305540235000 623360743125120 ... A008978;

6: 1 720 7484400 137225088000 3246670537110000 88832646059788350720 ... A008979;

with columns: A000142 (n=1), A000680 (n=2), A014606 (n=3), A014608 (n=4), A014609 (n=5).

A089759 is the transpose of this matrix. A034841 is its diagonal. A141906 is its lower triangle. A120666 is the upper triangle of this matrix with indices starting from 1. A248827 are the diagonal sums (or the row sums of the triangle).

(End)

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

Cf. A089759 (transposed), A141906 (subtriangle), A120666 (subtriangle transposed), A060538 (1st row/column removed).

Rows: A000012, A000984, A006480, A006480, A008978, A008979, A071549, A071550, A071551, A071552.

Columns: A000012, A000142, A000680, A014606, A014608, A014609, A248814, A172603, A172609, A172613.

Main diagonal gives: A034841.

Row sums of the triangle: A248827.

Programs

Magma

[Factorial(k*(n-k))/(Factorial(n-k))^k: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 26 2022

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