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A319251 Triangle read by rows: T(n,k) is the number of permutations pi of [n] with k descents such that s(pi) avoids the patterns 132, 231, 312, and 321, where s denotes West's stack-sorting map. 2
1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 28, 13, 1, 1, 19, 70, 70, 19, 1, 1, 26, 145, 250, 145, 26, 1, 1, 34, 266, 700, 700, 266, 34, 1, 1, 43, 448, 1666, 2548, 1666, 448, 43, 1, 1, 53, 708, 3528, 7644, 7644, 3528, 708, 53, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Rows are symmetric, so T(n,k) = T(n,n-1-k).
T(n,k) is the number of permutations of [n] that avoid the patterns 1342, 2341, 3142, 3241, 3412, and 3421 and have k descents.
LINKS
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
FORMULA
T(n,k) = N(n,k+1) + Sum_{i=1..n-2} Sum_{j=1..m} N(n-i-1,j) * N(i,k-j+1), where N(i,j) = (1/i) * binomial(i,j) * binomial(i,j-1) are the Narayana numbers given in A001263.
From Vladimir Kruchinin, Nov 16 2020: (Start)
G.f.: A001263(x,y) + x*A001263(x,y)^2.
T(n,k) = N(n,k) +2*C(n-1,k-2)*C(n-1,k)/(n-1). (End)
EXAMPLE
Triangle begins:
1,
1, 1,
1, 4, 1,
1, 8, 8, 1,
1, 13, 28, 13, 1,
1, 19, 70, 70, 19, 1
...
MATHEMATICA
Flatten[Table[Table[(1/n) Binomial[n, m + 1] Binomial[n, m] + Sum[Sum[(1/(n - i - 1)) Binomial[n - i - 1, j] Binomial[n - i - 1, j - 1] (1/i) Binomial[i, m - j + 1] Binomial[i, m - j], {j, 1, m}], {i, 1, n - 2}], {m, 0, n - 1}], {n, 1, 10}]]
CROSSREFS
Row sums give A071721. Cf. A001263, A319252.
Sequence in context: A347676 A177947 A132789 * A100754 A332307 A296405
KEYWORD
easy,nonn,tabl
AUTHOR
Colin Defant, Sep 15 2018
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)