A277493 Number of planar rooted Eulerian orientations (i.e., planar directed Eulerian maps with in-degree and out-degree equal for each vertex) with n edges.

1, 2, 10, 66, 504, 4216, 37548, 350090, 3380520, 33558024, 340670720, 3522993656, 37003723200, 393856445664, 4240313009272, 46109094112170, 505802627913960, 5591751626721960, 62247732179078880, 697269028520326328, 7854446696230804800, 88928907586405147872, 1011547523763310161600

Offset

0,2

Comments

a(n) is the coefficient of t^n in the generating function O(t) = sum_w O_w(t), where w is any binary word on the alphabet {0,1} having as many 0's as 1's.

The series O_w(t) are defined by O_epsilon(t) = 1 (for the empty word epsilon) and O_w(t) = t* sum_{w=aubv} O_u(t)*O_v(t) + t* sum_u O_uw'(t), where: a is a binary letter (0 or 1), b = 1-a, w' is the suffix of w of length |w|-1.

Links

Table of n, a(n) for n=0..22.

Nicolas Bonichon, Mireille Bousquet-Mélou, Paul Dorbec, and Claire Pennarun, On the number of planar Eulerian orientations, HAL preprint <hal-01389264> [math.CO], 2016.

Nicolas Bonichon, Mireille Bousquet-Mélou, Paul Dorbec, and Claire Pennarun, On the number of planar Eulerian orientations, arXiv:1610.09837 [math.CO], 2016.

Mireille Bousquet-Mélou and Andrew Elvey Price, The generating function of planar Eulerian orientations, arXiv:1803.08265 [math.CO], 2018.

Mireille Bousquet-Mélou, Andrew Elvey Price, and Paul Zinn-Justin, Eulerian orientations and the six-vertex model on planar map, arXiv:1902.07369 [math.CO], 2019. See Theorem 1.

Claire Pennarun, A C++ program generating the sequence

Andrew Elvey Price and Anthony J. Guttmann, Counting Planar Eulerian Orientations, arXiv:1707.09120 [math.CO], 2017

Formula

From Andrew Elvey Price, Dec 12 2024: (Start)

G.f.: (1/(2*t^2))*(t-R(t)) where R(t) is the g.f. of A324311.

a(n) = 2*A324312(n) for n>0. (End)

Crossrefs

Cf. A324312.

Sequence in context: A278460 A278462 A060206 * A361448 A376226 A346417

Adjacent sequences: A277490 A277491 A277492 * A277494 A277495 A277496

Keyword

nonn

Author

Claire Pennarun, Oct 17 2016

Extensions

a(16) onwards added by Andrew Howroyd, Dec 12 2024

Status

approved