A317555 Triangle read by rows: T(n,k) is the number of preimages of the permutation 21345...n under West's stack-sorting map that have k+1 valleys (1 <= k <= floor((n-1)/2)).
1, 4, 12, 2, 32, 16, 80, 80, 5, 192, 320, 60, 448, 1120, 420, 14, 1024, 3584, 2240, 224, 2304, 10752, 10080, 2016, 42, 5120, 30720, 40320, 13440, 840, 11264, 84480, 147840, 73920, 9240, 132, 24576, 225280, 506880, 354816, 73920, 3168
Offset: 3
Examples
Triangle begins:
1;
4;
12, 2;
32, 16;
80, 80, 5;
192, 320, 60;
448, 1120, 420, 14;
...
T(1,1) = 1 because the permutation 213 has one preimage under West's stack-sorting map (namely, 231), and this permutation has 2 valleys.
Links
- C. Defant, Preimages under the stack-sorting algorithm, arXiv:1511.05681 [math.CO], 2015-2018.
- C. Defant, Preimages under the stack-sorting algorithm, Graphs Combin., 33 (2017), 103-122.
- C. Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Crossrefs
Row sums give A002057.
Programs
-
Mathematica
Flatten[Table[Table[Sum[Sum[(2^(i - 2 j + 1)) Binomial[i - 1, 2 j - 2]CatalanNumber[j - 1] (2^((n - 1 - i) - 2 (m + 1 - j) + 1)) Binomial[(n - 1 - i) - 1, 2 (m + 1 - j) - 2] CatalanNumber[(m + 1 - j) - 1], {j, 1, m}], {i, 1, n - 2}], {m, 1, Floor[(n - 1)/2]}], {n, 1, 10}]]
Formula
T(n,k) = Sum_{i=1..n-2} Sum_{j=1..k} V(i,j) * V(n-1-i,m+1-j), where V(i,j) = 2^{i-2j+1} * (1/j) * binomial(i-1, 2j-2) * binomial(2j-2, j-1) are the numbers found in the triangle A091894.
Comments