A334394 Triangle read by rows: T(n,k) is the number of ordered triples of n-permutations with exactly k common descents, n>=0, 0<=k<=max(0,n-1).

1, 1, 7, 1, 163, 52, 1, 8983, 4499, 341, 1, 966751, 660746, 98256, 2246, 1, 179781181, 155729277, 35677082, 2045282, 15177, 1, 53090086057, 55690144728, 17446464519, 1754605504, 42658239, 104952, 1, 23402291822743, 28825420903351, 11518335730323, 1717307782339, 84058424389, 905365701, 739153, 1

Offset: 0

Author

Geoffrey Critzer, Apr 26 2020

Keywords

• nonn
• tabf

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.

Examples

			Triangle begins:
       1;
       1;
       7,      1;
     163,     52,     1;
    8983,   4499,   341,   1;
  966751, 660746, 98256, 2246, 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.

Crossrefs

Cf. A192721, A008292, A212856 (column k=0), A000442 (row sums).

Programs

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

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.