cp's OEIS Frontend

A090441 Symmetric triangle of certain normalized products of decreasing factorials.

%I A090441 #31 Jun 02 2025 16:49:53
%S A090441 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,
%T A090441 2880,120,1,1,720,86400,1036800,1036800,86400,720,1,1,5040,3628800,
%U A090441 217728000,870912000,217728000,3628800,5040,1,1,40320,203212800,73156608000
%N A090441 Symmetric triangle of certain normalized products of decreasing factorials.
%C A090441 Similar to, but different from, superfactorial Pascal triangle A009963.
%C A090441 A009963(n,m) = (Product_{p=0..m-1} (n-p)!)/superfac(m) with n >= m >= 0, otherwise 0.
%C A090441 From _Natalia L. Skirrow_, Apr 13 2025 (Start)
%C A090441 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.
%C A090441 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.
%C A090441 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)).
%C A090441 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).
%C A090441 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).
%C A090441 (End)
%H A090441 Donald Knuth, <a href="https://arxiv.org/abs/math/9205211">Two Notes on Notation</a>, 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.)
%H A090441 Wolfdieter Lang, <a href="/A090441/a090441_1.txt">First 9 rows</a>
%H A090441 Richard P. Stanley, <a href="https://math.mit.edu/~rstan/pubs/pubfiles/9.pdf">Ordered Structures and Partitions</a>, 1971.
%F A090441 a(n, m) = 0 if n < m;
%F A090441 a(n, m) = 1 if m = 0 or m = n;
%F A090441 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).
%F A090441 a(n, m) = superfac(n-1)/superfac(m-1)/superfac(n-m-1)
%F A090441 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
%e A090441 Rows for n = 0, 1, 2, 3, ...:
%e A090441   1;
%e A090441   1,  1;
%e A090441   1,  1,  1;
%e A090441   1,  2,  2,  1;
%e A090441   1,  6, 12,  6,  1;
%e A090441   ...
%o A090441 (PARI) spf(n) = prod(k=2, n, k!);
%o A090441 T(n,m) = spf(n-1)/spf(m-1)/spf(n-m-1);
%o A090441 row(n) = vector(n+1, k, T(n, k-1)); \\ _Michel Marcus_, Apr 13 2025
%Y A090441 Column sequences give: A000012 (powers of 1), A000142 (factorials), A010790, A090443-4, etc.
%Y A090441 Cf. A090445 (row sums), A090446 (alternating row sums).
%K A090441 nonn,easy,tabl
%O A090441 0,8
%A A090441 _Wolfdieter Lang_, Dec 23 2003
%E A090441 OFFSET changed from -1 to 0 by _Natalia L. Skirrow_, Apr 13 2025