A334257 Triangle read by rows: T(n,k) is the number of ordered pairs of n-permutations with exactly k common double descents, n>=0, 0<=k<=max{0,n-2}.

1, 1, 4, 35, 1, 545, 30, 1, 13250, 1101, 48, 1, 463899, 51474, 2956, 70, 1, 22106253, 3070434, 217271, 7545, 96, 1, 1375915620, 229528818, 19372881, 864632, 20322, 126, 1, 108386009099, 21107789247, 2070917370, 113587335, 3530099, 61089, 160, 1

Offset: 0

Author

Geoffrey Critzer, Apr 26 2020

Keywords

• nonn
• tabf

Comments

An ordered pair of n-permutations ((a_1,a_2,...,a_n),(b_1,b_2,...,b_n)) has a common double descent at position i, 1<=i<=n-2, if a_i > a_i+1 > a_i+2 and b_i > b_i+1 > b_i+2.

Examples

			T(4,1) = 30:  There are 9 such ordered pairs formed from the permutations 3421,2431,1432.  There are 9 such ordered pairs formed from the permutations 4312,4213,3214.  Then pairing each of these 6 permutations with 4321 gives 12 more ordered pairs with exactly 1 common double descent.  9+9+12 = 30.
Triangle T(n,k) begins:
       1;
       1;
       4;
      35,     1;
     545,    30,    1;
   13250,  1101,   48,  1;
  463899, 51474, 2956, 70, 1;
  ...
		

References

• R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, example 3.18.3e, page 366.

Links

• Alois P. Heinz, Rows n = 0..60, flattened
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 209.

Crossrefs

Column k=0 gives A334412.

Cf. A192721, A162975.

Programs

Maple

b:= proc(n, u, v, t) option remember; expand(`if`(n=0, 1,
      add(add(b(n-1, u-j, v-i, x)*t, i=1..v)+
          add(b(n-1, u-j, v+i-1, 1), i=1..n-v), j=1..u)+
      add(add(b(n-1, u+j-1, v-i, 1), i=1..v)+
          add(b(n-1, u+j-1, v+i-1, 1), i=1..n-v), j=1..n-u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2, 1)):
seq(T(n), n=0..10);  # Alois P. Heinz, Apr 26 2020

Mathematica

nn = 8; a = Apply[Plus,Table[Normal[Series[y x^3/(1 - y x - y x^2), {x, 0, nn}]][[n]]/(n +2)!^2, {n, 1, nn - 2}]] /. y -> y - 1; Map[Select[#, # > 0 &] &,
  Range[0, nn]!^2 CoefficientList[Series[1/(1 - x - a), {x, 0, nn}], {x, y}]] // Grid