A383372 Number of centrally symmetric Baxter permutations of length n.

1, 1, 2, 2, 6, 8, 26, 38, 130, 202, 712, 1152, 4144, 6904, 25202, 42926, 158442, 274586, 1022348, 1796636, 6736180, 11974360, 45154320, 81040720, 307069360, 555620080, 2113890560, 3851817920, 14705955008, 26960013552, 103245460226

Offset

0,3

Comments

For all n > 0, a(n) is the number of triples of non-intersecting lattice paths of length n-1.

a(n) is the number of symmetric twin pairs of full binary trees with n internal nodes.

Links

Table of n, a(n) for n=0..30.

Kevin Dilks, Involutions on Baxter Objects, arXiv:1402.2961 [math.CO], 2014.

Stefan Felsner, Éric Fusy, Marc Noy, and David Orden, Bijections for Baxter families and related objects, J. Combin. Theory Ser. A, 118(3):993-1020, 2011.

Formula

For all n>0, a(n) = Sum_{k=0...n-1} Theta_{k,n-k-1}, where Theta_{k,l} is equal to:

- C(a+b+1,a+1)*C(a+b+1,a)*C(a+b,a)/(a+b+1) if k and l are even with k = 2*a and l = 2*b;

- C(a+b+1,a+1)^2*C(a+b+1,a)/(a+b+1) if k is odd and l is even with k = 2*a+1 and l = 2*b;

- Theta(l,k) if k is even and l is odd;

- 0 if k and l are odd.

Example

The Baxter permutations corresponding to a(4) = 6 are 1234, 1324, 2143, 3412, 4231, and 4321.

Crossrefs

Cf. A001181.

Sequence in context: A275441 A197465 A074933 * A157253 A003178 A079494

Adjacent sequences: A383369 A383370 A383371 * A383373 A383374 A383375

Keyword

nonn

Author

Ludovic Schwob, Apr 24 2025

Status

approved