cp's OEIS Frontend

A358177 Number of Eulerian orientations of a (labeled) 2n-dimensional hypercube graph, Q_2n. Q_2n is also the n-dimensional torus grid graph (C_4)^n.

%I A358177 #30 Feb 16 2025 08:34:04
%S A358177 1,2,2970,351135773356461511142023680
%N A358177 Number of Eulerian orientations of a (labeled) 2n-dimensional hypercube graph, Q_2n. Q_2n is also the n-dimensional torus grid graph (C_4)^n.
%C A358177 An Eulerian orientation of a graph is an orientation of the edges such that every vertex has in-degree equal to out-degree. (C_4)^n denotes the Cartesian product of n cycle graphs on 4 nodes.
%H A358177 Mikhail Isaev, Brendan D. McKay, and Rui-Ray Zhang, <a href="https://arxiv.org/abs/2409.04989">Correlation between residual entropy and spanning tree entropy of ice-type models on graphs</a>, arXiv:2409.04989 [math.CO], 2024. See p. 24.
%H A358177 A. Schrijver, <a href="https://ir.cwi.nl/pub/10053">Bounds on the Number of Eulerian Orientations</a>, Combinatorica, 3 (3--4), (1983), 375-380.
%H A358177 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCartesianProduct.html">Cartesian product of graphs</a>
%H A358177 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HypercubeGraph.html">Hypercube Graph</a>
%F A358177 a(0) = A007081(2^0) = 1.
%F A358177 a(1) = A334553(1) = 2.
%F A358177 a(2) = A054759(4) = 2970.
%F A358177 Schrijver (1983) provides general bounds on unknown terms of the form (2^(-k) * binomial(2k,k))^(2^(2k)) <= a(k) <= sqrt(binomial(2k,k)^(2^(2k))).
%F A358177 From this we have the specific bounds 2.9*10^25 <= a(3) <= 4.3*10^41 and 1.2*10^164 <= a(4) <= 1.5*10^236.
%e A358177 For n = 1, dimension 2n = 2, there are two Eulerian orientations (the cyclic ones). So a(1) = 2.
%Y A358177 Cf. A007081, A054759, A298119, A307334, A334553.
%K A358177 nonn,hard,more
%O A358177 0,2
%A A358177 _Peter Munn_ and _Zachary DeStefano_, Nov 02 2022
%E A358177 a(3) added by _Brendan McKay_, Nov 04 2022