OFFSET
0,3
COMMENTS
An ordered triple of n-permutations ( (a_1,a_2,...,a_n),(b_1,b_2,...,b_n),(c_1,c_2,...,c_n) ) has a common descent at position i, 1<=i<=n-1, if a_i > a_i+1, b_i > b_i+1 and c_i > c_i+1.
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, example 3.18.3e, page 366.
LINKS
Alois P. Heinz, Rows n = 0..30, flattened
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 209.
FORMULA
Sum_{n>=0} Sum_{k>=0} T(n,k)*y^k*x^n/n!^3 = (y-1)/(y-f(x*(y-1))) where f(z) = Sum_{n>=0} z^n/n!^3.
EXAMPLE
Triangle begins:
1;
1;
7, 1;
163, 52, 1;
8983, 4499, 341, 1;
966751, 660746, 98256, 2246, 1;
...
MAPLE
T:= (n, k)-> n!^3*coeff(series(coeff(series((y-1)/(y-add((x*
(y-1))^j/j!^3, j=0..n)), y, k+1), y, k), x, n+1), x, n):
seq(seq(T(n, k), k=0..max(0, n-1)), n=0..10); # Alois P. Heinz, Apr 28 2020
MATHEMATICA
nn = 6; e3[x_] := Sum[x^n/n!^3, {n, 0, nn}]; Drop[Map[Select[#, # > 0 &] &,
Table[n!^3, {n, 0, nn}] CoefficientList[Series[(y - 1)/(y - e3[x (y - 1)]), {x, 0, nn}], {x, y}]], 1] // Grid
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Apr 26 2020
STATUS
approved