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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, 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 (list; table; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,tabl
AUTHOR
Tian Han, Nov 24 2023
STATUS
approved

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Last modified July 12 16:40 EDT 2024. Contains 374251 sequences. (Running on oeis4.)