Séminaire Lotharingien de Combinatoire, 82B.70 (2019), 12 pp.

Mireille Bousquet-Mélou, Andrew Elvey Price, and Paul Zinn-Justin

Eulerian orientations and the six-vertex model on planar maps

Abstract. We address the enumeration of planar 4-valent maps equipped with an Eulerian orientation by two different methods, and compare the solutions we thus obtain. With the first method we enumerate these orientations as well as a restricted class which we show to be in bijection with general Eulerian orientations. The second method, based on the work of Kostov, allows us to enumerate these 4-valent orientations with a weight on some vertices, corresponding to the six vertex model. We prove that this result generalises both results obtained using the first method, although the equivalence is not immediately clear.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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