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

Author

Ludovic Schwob, Apr 24 2025

Keywords

• nonn

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.

Examples

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

Links

• Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, Bijections for Baxter families and related objects, J. Combin. Theory Ser. A, 118(3):993-1020, 2011.
• Kevin Dilks, Involutions on Baxter Objects, arXiv:1402.2961 [math.CO], 2014.

Crossrefs

Cf. A001181.

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.