A185249 Triangle read by rows: Table III.5 of Myriam de Sainte-Catherine's 1983 thesis.

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 5, 0, 1, 0, 3, 0, 14, 0, 1, 1, 0, 14, 0, 42, 0, 1, 0, 4, 0, 84, 0, 132, 0, 1, 1, 0, 30, 0, 594, 0, 429, 0, 1, 0, 5, 0, 330, 0, 4719, 0, 1430, 0, 1, 1, 0, 55, 0, 4719, 0, 40898, 0, 4862, 0, 1, 0, 6, 0, 1001, 0, 81796, 0, 379236, 0, 16796, 0, 1, 1, 0, 91, 0, 26026, 0, 1643356, 0, 3711916, 0, 58786, 0, 1

Offset: 0

Author

N. J. A. Sloane, Feb 15 2011

Keywords

• tabl
• nonn

Comments

I have a photocopy of certain pages of the thesis, but unfortunately not enough to find the definition of this table. I have written to the author.

(Added later) However, Alois P. Heinz found a formula involving Catalan numbers which matches all the data and is surely correct, so the triangle is no longer a mystery.

Reading upwards along antidiagonals gives A123352.

From Petros Hadjicostas, Sep 04 2019: (Start)

Consider "Young tableaux with entries from the set {1,...,n}, strictly increasing in rows and not decreasing in columns. Note that usually the reverse convention between rows and columns is used."

de Sainte-Catherine and Viennot (1986) proved that "the number b_{n,k} of such Young tableaux having only columns with an even number of elements and bounded by height p = 2*k" is given by b_{n,k} = Product_{1 <= i <= j <= n} (2*k + i + j)/(i + j)." In Section 6 of their paper, they give an interpretation of this formula in terms of Pfaffians and perfect matchings.

It turns out that for the current array, T(n,k) = b_{k, (n-k)/2} if n-k is even, and 0 otherwise (for n >= 0 and 0 <= k <= n). It is unknown, however, what kind of interpretation Myriam de Sainte-Catherine gave to the number T(n,k) three years earlier in her 1983 Ph.D. dissertation. It may be distantly related to the numbers b_{n,k} that are found in her 1986 paper with G. Viennot.

(End)

The T(n, k) for n and k same parity are the numbers in the upper triangle of the Catalan Number Wall in "Number Walls in Combinatorics". Thus 0 = T(n-1, k+1)*T(n+1, k-1) - T(n-1, k-1)*T(n+1, k+1) + T(n, k)^2 for all n, k. - Michael Somos, Aug 15 2023

Examples

			Triangle begins:
  1
  0 1
  1 0  1
  0 2  0    1
  1 0  5    0     1
  0 3  0   14     0     1
  1 0 14    0    42     0       1
  0 4  0   84     0   132       0      1
  1 0 30    0   594     0     429      0       1
  0 5  0  330     0  4719       0   1430       0     1
  1 0 55    0  4719     0   40898      0    4862     0     1
  0 6  0 1001     0 81796       0 379236       0 16796     0 1
  1 0 91    0 26026     0 1643356      0 3711916     0 58786 0 1
  ...
		

References

• Myriam de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire, Physique et Informatique. Ph.D. Dissertation, Université Bordeaux I, 1983.

Links

• Alois P. Heinz, Rows n = 0..100, flattened
• M. de Sainte-Catherine and G. Viennot, Enumeration of certain Young tableaux with bounded height, in: G. Labelle and P. Leroux (eds), Combinatoire énumérative, Lecture Notes in Mathematics, vol. 1234, Springer, Berlin, Heidelberg, 1986, pp. 58-67.
• M. Somos, Number Walls in Combinatorics, 2000.

Crossrefs

Diagonals give A000108, A005700, A006149, A006150, A006151, A000330, A006858, A091962.

Row sums give A186232. Nonzero diagonals give columns in A078920.

Cf. A179898.

Programs

Maple

with(LinearAlgebra):
ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
T := proc(n, k)
       if n=k then 1
     elif irem(n+k, 2)=1 then 0
     else Determinant(Matrix((n-k)/2, (i, j)-> ctln(i+j-1+k)))
       fi
     end:
seq(seq(T(n,k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 15 2011

Mathematica

t[n_, n_] = 1; t[n_, k_] /; Mod[n+k, 2] == 1 = 0; t[n_, k_] := Array[CatalanNumber[#1 + #2 - 1 + k]&, {(n-k)/2, (n-k)/2}] // Det; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014, after Alois P. Heinz *)

PARI

{T(n, k) = if((n-k)%2||k<0||k>n, 0, prod(i=1, k, prod(j=i, k, (n-k+i+j)/(i+j))))}; /* Michael Somos, Aug 15 2023 */

Formula

T(n,k) = Product_{1 <= i <= j <= k} (n-k + i + j)/(i + j) if n - k is even, and = 0 otherwise (for n >= 0 and 0 <= k <= n). - Petros Hadjicostas, Sep 04 2019

Extensions

Typo in data corrected by Alois P. Heinz, Feb 15 2011