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.

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, 1092

Offset

1,7

Comments

Rows are symmetric: T(n,k) = T(n,2n-k).

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.

Row sums give A180874.

Links

Table of n, a(n) for n=1..27.

Colin Defant, Michael Engen, and Jordan A. Miller, Stack-sorting, set partitions, and Lassalle's sequence, arXiv:1809.01340 [math.CO], 2018.

Formula

T(n,1) = T(n,2n-1) = 0 for n>1.

T(n,2) = T(n,2n-2) = A180874(n-1) for n>1.

Example

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.
Triangle begins:
             1,
         0,  1,  0,
     0,  1,  3,  1, 0,
  0, 5, 13, 20, 13, 5, 0,
  ...

Crossrefs

Cf. A180874.

Sequence in context: A318502 A115090 A112295 * A110517 A091925 A369199

Adjacent sequences: A318970 A318971 A318972 * A318974 A318975 A318976

Keyword

nonn,tabf,more

Author

Colin Defant, Sep 06 2018

Status

approved