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User: Tian Han

Tian Han has authored 2 sequences.

A367873 Irregular triangle read by rows: T(n,k) = number of permutations of length n avoiding simultaneously the patterns 132 and 321 with the maximum number of non-overlapping ascents equal to k.

1, 1, 1, 0, 4, 0, 4, 3, 0, 0, 11, 0, 0, 9, 7, 0, 0, 0, 22, 0, 0, 0, 16, 13, 0, 0, 0, 0, 37, 0, 0, 0, 0, 25, 21, 0, 0, 0, 0, 0, 56, 0, 0, 0, 0, 0, 36, 31, 0, 0, 0, 0, 0, 0, 79, 0, 0, 0, 0, 0, 0, 49, 43, 0, 0, 0, 0, 0, 0, 0, 106

Offset: 0

Tian Han, Dec 03 2023

An ascent in a permutation a(1)a(2)...a(n) is position i such that a(i) < a(i+1).

			1,
1, 1,
0, 4,
0, 4, 3,
0, 0, 11,
0, 0, 9, 7,
0, 0, 0, 22,
0, 0, 0, 16, 13,
0, 0, 0, 0, 37,
0, 0, 0, 0, 25, 21,
0, 0, 0, 0, 0, 56,
0, 0, 0, 0, 0, 36, 31,
0, 0, 0, 0, 0, 0, 79,
0, 0, 0, 0, 0, 0, 49, 43,
0, 0, 0, 0, 0, 0, 0, 106

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

Cf. A367631.

G.f.: (1 + x + x^2 - 2*x^2*y + x^3*y + x^4*y + 3*x^4*y^2 + 2*x^5*y^2)/(1 - x^2*y)^3.

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.

Original entry on oeis.org

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

Tian Han, Nov 24 2023

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).

			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;
  ...

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.

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).

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

cp's OEIS Frontend

User: Tian Han

A367873 Irregular triangle read by rows: T(n,k) = number of permutations of length n avoiding simultaneously the patterns 132 and 321 with the maximum number of non-overlapping ascents equal to k.

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