|
%I #14 Sep 15 2018 05:26:00
%S 1,4,12,2,32,16,80,80,5,192,320,60,448,1120,420,14,1024,3584,2240,224,
%T 2304,10752,10080,2016,42,5120,30720,40320,13440,840,11264,84480,
%U 147840,73920,9240,132,24576,225280,506880,354816,73920,3168
%N 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)).
%C If pi is any permutation of [n] with exactly 1 descent, then the number of preimages of pi under West's stack-sorting map that have k+1 valleys is at most T(n,k).
%H C. Defant, <a href="https://arxiv.org/abs/1511.05681">Preimages under the stack-sorting algorithm</a>, arXiv:1511.05681 [math.CO], 2015-2018.
%H C. Defant, <a href="https://doi.org/10.1007/s00373-016-1752-5">Preimages under the stack-sorting algorithm</a>, Graphs Combin., 33 (2017), 103-122.
%H C. Defant, <a href="https://arxiv.org/abs/1809.03123">Stack-sorting preimages of permutation classes</a>, arXiv:1809.03123 [math.CO], 2018.
%F 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.
%e Triangle begins:
%e 1;
%e 4;
%e 12, 2;
%e 32, 16;
%e 80, 80, 5;
%e 192, 320, 60;
%e 448, 1120, 420, 14;
%e ...
%e 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.
%t 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}]]
%Y Row sums give A002057.
%K easy,nonn,tabf
%O 3,2
%A _Colin Defant_, Sep 14 2018
|
