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

Author

Colin Defant, Sep 06 2018

Keywords

• nonn
• tabf
• more

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.

Examples

			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,
  ...
		

Links

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

Crossrefs

Cf. A180874.

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.