cp's OEIS Frontend

A383372 Number of centrally symmetric Baxter permutations of length n.

%I A383372 #7 Apr 25 2025 12:29:41
%S A383372 1,1,2,2,6,8,26,38,130,202,712,1152,4144,6904,25202,42926,158442,
%T A383372 274586,1022348,1796636,6736180,11974360,45154320,81040720,307069360,
%U A383372 555620080,2113890560,3851817920,14705955008,26960013552,103245460226
%N A383372 Number of centrally symmetric Baxter permutations of length n.
%C A383372 For all n > 0, a(n) is the number of triples of non-intersecting lattice paths of length n-1.
%C A383372 a(n) is the number of symmetric twin pairs of full binary trees with n internal nodes.
%H A383372 Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, <a href="https://www.sciencedirect.com/science/article/pii/S0097316510000671">Bijections for Baxter families and related objects</a>, J. Combin. Theory Ser. A, 118(3):993-1020, 2011.
%H A383372 Kevin Dilks, <a href="https://arxiv.org/abs/1402.2961">Involutions on Baxter Objects</a>, arXiv:1402.2961 [math.CO], 2014.
%F A383372 For all n>0, a(n) = Sum_{k=0...n-1} Theta_{k,n-k-1}, where Theta_{k,l} is equal to:
%F A383372   - 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;
%F A383372   - 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;
%F A383372   - Theta(l,k) if k is even and l is odd;
%F A383372   - 0 if k and l are odd.
%e A383372 The Baxter permutations corresponding to a(4) = 6 are 1234, 1324, 2143, 3412, 4231, and 4321.
%Y A383372 Cf. A001181.
%K A383372 nonn
%O A383372 0,3
%A A383372 _Ludovic Schwob_, Apr 24 2025