A367631 Triangle read by rows: T(n,k) is the number of permutations of length n avoiding simultaneously the patterns 123 and 132 with the maximum number of non-overlapping descents equal k.

1, 1, 0, 1, 1, 0, 0, 4, 0, 0, 0, 5, 3, 0, 0, 0, 2, 14, 0, 0, 0, 0, 0, 23, 9, 0, 0, 0, 0, 0, 16, 48, 0, 0, 0, 0, 0, 0, 4, 97, 27, 0, 0, 0, 0, 0, 0, 0, 94, 162, 0, 0, 0, 0, 0, 0, 0, 0, 44, 387, 81, 0, 0, 0, 0, 0, 0, 0, 0, 8, 476, 540, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 320, 1485, 243, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Tian Han, Nov 24 2023

Keywords

• nonn
• tabl

Comments

Number of permutations of length n avoiding simultaneously the patterns 123 and 132 with the maximum number of non-overlapping descents equal k. A descent in a permutation a(1)a(2)...a(n) is position i such that a(i) > a(i+1).

Examples

			Triangle T(n,k) begins:
  1;
  1, 0;
  1, 1,  0;
  0, 4,  0,  0;
  0, 5,  3,  0,   0;
  0, 2, 14,  0,   0,    0;
  0, 0, 23,  9,   0,    0,   0;
  0, 0, 16, 48,   0,    0,   0, 0;
  0, 0,  4, 97,  27,    0,   0, 0, 0;
  0, 0,  0, 94, 162,    0,   0, 0, 0, 0;
  0, 0,  0, 44, 387,   81,   0, 0, 0, 0, 0;
  0, 0,  0,  8, 476,  540,   0, 0, 0, 0, 0, 0;
  0, 0,  0,  0, 320, 1485, 243, 0, 0, 0, 0, 0, 0;
  ...
		

Links

• Tian Han and Sergey Kitaev, Joint distributions of statistics over permutations avoiding two patterns of length 3, arXiv:2311.02974 [math.CO], 2023. See formula 7 at page 7.

Crossrefs

Row sums give A011782.

Column sums give 3*A005054.

T(2n,n) gives A133494.

T(3n+2,n) gives A000079.

T(3n+1,n) gives A053220(n+1).

Formula

G.f.: (1 + x + x^2 - 2*x^2*z - x^3*z)/(1 - 3*x^2*z - 2*x^3*z).