%I #14 Sep 15 2018 05:26:17
%S 1,0,1,0,0,1,3,1,0,0,5,13,20,13,5,0,0,56,136,221,266,221,136,56,0,0,
%T 1092
%N Triangle read by rows: T(n,k) is the number of permutations of [2n-1] that have exactly one preimage under West's stack-sorting map and that also have first entry k.
%C Rows are symmetric: T(n,k) = T(n,2n-k).
%C It appears that the sequence T(n,1),...,T(n,2n-1) is always unimodal. In fact, it appears that this sequence is always log-concave.
%C Row sums give A180874.
%H Colin Defant, Michael Engen, and Jordan A. Miller, <a href="https://arxiv.org/abs/1809.01340">Stack-sorting, set partitions, and Lassalle's sequence</a>, arXiv:1809.01340 [math.CO], 2018.
%F T(n,1) = T(n,2n-1) = 0 for n>1.
%F T(n,2) = T(n,2n-2) = A180874(n-1) for n>1.
%e The five uniquely sorted permutations of [5] are 21435, 31425, 32415, 32145, and 42135. Of these permutations, T(3,1) = 0 start with the entry 1, T(3,2) = 1 starts with 2, T(3,3) = 3 start with 3, T(3,4) = 1 starts with 4, and T(3,5) = 0 start with 5.
%e Triangle begins:
%e 1,
%e 0, 1, 0,
%e 0, 1, 3, 1, 0,
%e 0, 5, 13, 20, 13, 5, 0,
%e ...
%Y Cf. A180874.
%K nonn,tabf,more
%O 1,7
%A _Colin Defant_, Sep 06 2018