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Bijective counting of plane bipolar orientations and Schnyder woods
 European J. Combin
"... Abstract. A bijection Φ is presented between plane bipolar orientations with prescribed numbers of vertices and faces, and nonintersecting triples of upright lattice paths with prescribed extremities. This yields a combinatorial proof of the following formula due to R. Baxter for the number Θij of ..."
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Abstract. A bijection Φ is presented between plane bipolar orientations with prescribed numbers of vertices and faces, and nonintersecting triples of upright lattice paths with prescribed extremities. This yields a combinatorial proof of the following formula due to R. Baxter for the number Θij of plane bipolar orientations with i nonpolar vertices and j inner faces:
BIJECTIVE COUNTING OF PLANE BIPOLAR ORIENTATIONS
"... Abstract. We introduce a bijection between plane bipolar orientations with fixed numbers of vertices and faces, and nonintersecting triples of upright lattice paths with some specific extremities. Writing ϑij for the number of plane bipolar orientations with (i + 1) vertices and (j +1) faces, our b ..."
Abstract

Cited by 1 (1 self)
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Abstract. We introduce a bijection between plane bipolar orientations with fixed numbers of vertices and faces, and nonintersecting triples of upright lattice paths with some specific extremities. Writing ϑij for the number of plane bipolar orientations with (i + 1) vertices and (j +1) faces, our bijection provides a combinatorial proof of the following formula due to Baxter: (1) ϑij = 2 (i + j − 2)! (i + j − 1)! (i + j)!
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"... We introduce a bijection between plane bipolar orientations with fixed numbers of vertices and faces, and nonintersecting triples of upright lattice paths with some specific extremities. Writing ϑij for the number of plane bipolar orientations with (i+1) vertices and (j +1) faces, our bijection pro ..."
Abstract
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We introduce a bijection between plane bipolar orientations with fixed numbers of vertices and faces, and nonintersecting triples of upright lattice paths with some specific extremities. Writing ϑij for the number of plane bipolar orientations with (i+1) vertices and (j +1) faces, our bijection provides a combinatorial proof of the following formula due to Baxter: (1) ϑij = 2 (i + j − 2)! (i + j − 1)! (i + j)!