A318973 - OEIS

A318973

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.

%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