Abstract
Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations. The Chan-Robbins-Yuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which M\'esz\'aros gave an interpretation as the volume of a collection of flow polytopes. We introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes. Our results generalize M\'esz\'aros's construction and a recent flow polytope interpretation of the Morris identity by Corteel-Kim-M\'esz\'aros. We prove the product formula of our refinement following the strategy of the Baldoni-Vergne proof of the Morris identity. Lastly, we study a symmetry of the Morris identity bijectively using the Danilov-Karzanov-Koshevoy triangulation of flow polytopes and a bijection of M\'esz\'aros-Morales-Striker.
Highlights
Foreword Flow polytopes play a fundamental role in combinatorial optimization through their relation to maximum matching and minimum cost problems
In analogy to the Catalan numbers, the Narayana number N (n, k) counts the number of lattice paths from (0, 0) toC.(Rn. ,Mna)thétmhaattiqudeo—n2o0t21p, 3a5s9s, na 7b,o8v23e-8t5h1e line y = x and has 2k − 1 turns. Both Narayana and Catalan numbers appear in Theorem 1.5, where the Narayana refine the volume of the CRY polytope
This paper provides multiple combinatorial proofs of recurrence relations for Ψn(k, a, b, c) that could contribute to a combinatorial proof of the Morris constant term identity, and the volume formula for the Chan–Robbins–Yuen polytope
Summary
The following theorem, which appears in unpublished work of Postnikov and Stanley and in the work of Baldoni–Vergne [2], relates the volume of a flow polytope to a Kostant partition function. Corteel–Kim–Mészáros [7, Theorem 1.2] showed that for any positive a, b, and c, Mn(a, b, c) gives the volume of the flow polytope on the following graph. REFINEMENTS AND SYMMETRIES OF THE MORRIS IDENTITY FOR VOLUMES OF FLOW POLYTOPES kna,+b,2c a a 0 c b n+1 b c a − 1 + c(i − 1). We compute the following explicit product formula for Ψn(k, a, b, c) that completes our refinement and new proof of the Morris identity. We leave as an open problem to prove this relation combinatorially, which would imply a combinatorial proof of the volume formula for the CRY polytope (Theorem 3). The bijection holds for any graph G and as a special case we obtain a bijection of Postnikov [27] between lattice points of (p − 1)∆q−1 and (q − 1)∆p−1 further studied in [12]
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