A090441 Symmetric triangle of certain normalized products of decreasing factorials.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 12, 6, 1, 1, 24, 144, 144, 24, 1, 1, 120, 2880, 8640, 2880, 120, 1, 1, 720, 86400, 1036800, 1036800, 86400, 720, 1, 1, 5040, 3628800, 217728000, 870912000, 217728000, 3628800, 5040, 1, 1, 40320, 203212800, 73156608000

Offset: 0

Author

Wolfdieter Lang, Dec 23 2003

Keywords

• nonn
• easy
• tabl

Comments

Similar to, but different from, superfactorial Pascal triangle A009963.

A009963(n,m) = (Product_{p=0..m-1} (n-p)!)/superfac(m) with n >= m >= 0, otherwise 0.

From Natalia L. Skirrow, Apr 13 2025 (Start)

Denoting this sequence as the superbinomial sb(n,k), the hook length formula for a j X k rectangular Young tableau states the number of configurations of j*k distinct numbers such that each row and column is strictly increasing is (j*k)!/sb(j+k,j), ie. 1/sb(j+k,j) is the probability that a random permutation is a Young tableau.

Meanwhile, if the numbers are placed into the array with repetition, but the columns are still strictly increasing, there are c(n,j,k) = sb(n+1,j+k)/(sb(n+1-j,k)*sb(n+1-k,j)) configurations.

If the strict criterion is relaxed to monotonic, this becomes C(n,j,k) = sb(n-1+j+k,j+k)/(sb(n-1+j,j)*sb(n-1+k,k)).

By proposition 13.2(i) of Stanley's PhD thesis, for fixed j,k, c(n,j,k) and C(n,j,k) are polynomials in n of degree j*k, and c(n,j,k) = (-1)^(j*k)*C(-n,j,k).

For example, c(n,1,k)=(n choose k) and C(n,1,k)=(n+k-1 choose k), while c(n,2,k) = N(n,k+1) and C(n,2,k) = N(n+k,k+1), so the binomial coefficients and Narayana numbers N=A001263 obey the dualities (under continuation as polynomials) (n choose k) = (-1)^k*(k-1-n choose k) and N(n,k) = N(k-1-n,k).

(End)

Examples

			Rows for n = 0, 1, 2, 3, ...:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  2,  1;
  1,  6, 12,  6,  1;
  ...
		

Links

• Donald Knuth, Two Notes on Notation, arXiv:math/9205211 [math.HO], 1992. (Page 16-17 explain and give examples; the case with Narayana numbers come from tying together the poset P_k's 'shoelaces' with inequalities, into a 2 X k rectangle.)
• Wolfdieter Lang, First 9 rows
• Richard P. Stanley, Ordered Structures and Partitions, 1971.

Crossrefs

Column sequences give: A000012 (powers of 1), A000142 (factorials), A010790, A090443-4, etc.

Cf. A090445 (row sums), A090446 (alternating row sums).

Programs

PARI

spf(n) = prod(k=2, n, k!);
T(n,m) = spf(n-1)/spf(m-1)/spf(n-m-1);
row(n) = vector(n+1, k, T(n, k-1)); \\ Michel Marcus, Apr 13 2025

Formula

a(n, m) = 0 if n < m;

a(n, m) = 1 if m = 0 or m = n;

a(n, m) = (Product_{p=1..m} (n-p)!)/superfac(m-1) if n >= 0, 1 <= m <= n+1, where superfac(n) := A000178(n), n >= 0, (superfactorials).

a(n, m) = superfac(n-1)/superfac(m-1)/superfac(n-m-1)

With offset 1, equals ConvOffsStoT transform of the factorials, A000142: (1, 1, 2, 6, 24, ...); e.g., ConvOffs transform of (1, 1, 2, 6) = (1, 6, 12, 6, 1). - Gary W. Adamson, Apr 21 2008

Extensions

OFFSET changed from -1 to 0 by Natalia L. Skirrow, Apr 13 2025