|
| |
|
|
A185249
|
|
Triangle read by rows: Table III.5 of Myriam de Sainte-Catherine's 1983 thesis.
|
|
6
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,8
|
|
|
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.
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
|
|
|
REFERENCES
|
Myriam de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire. Physique et Informatique. Ph.D. Dissertation, Université Bordeaux I, 1983.
|
|
|
LINKS
|
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.
|
|
|
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
|
|
|
EXAMPLE
|
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
...
|
|
|
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:
|
|
|
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 *)
|
|
|
PROG
|
(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 */
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
