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

0,8

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).

Links

Table of n, a(n) for n=0..90.

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.

Formula

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

Example

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;
  ...

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).

Sequence in context: A071326 A284103 A151674 * A297968 A243000 A285214

Adjacent sequences: A367628 A367629 A367630 * A367632 A367633 A367634

Keyword

nonn,tabl

Author

Tian Han, Nov 24 2023

Status

approved