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While working with Grothendieck's Quot-schemes and their virtual enumerative invariants, the author came across a combinatorial identity which would have followed as a consequence of a very general abstract machinery. We dedicate this short note to give a combinatorial proof of this identity which is closely related to the Lagrange inversion. A special case of the result was proved by Mathoverflow users "Alex Gavrilov" and "esg" answering our enquiry.

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We study punctual Quot-schemes of projective curves, surfaces and Calabi-Yau fourfolds and the symmetry of their (virtual) invariants. Let Y be a variety in the above three classes, E Y → Y a torsion-free sheaf and Quot Y (E Y , n) the quot-scheme. Under some assumptions in the 4-fold case, Quot Y (E Y , n) is projective and carries a virtual fundamental class. Our goal is to give a(n almost) complete description of these classes and tautological integrals over them.We use novel methods developed in [8] relying on the wall-crossing framework of Joyce whichhave not yet been applied to this setting. We summarize here the main results:1. We show that the virtual cobordism class of Quot S (E S , n) depends only on rank. This implies in particular that for computing virtual Euler characteristic we may reduced to the case E S = C e .2. The quotient of generating series of tautological invariants is expressed in terms of a universal power-series and c 1 (E S ). This leads to a generalization of rationality statements where we additional determine the poles for the χ y -genus.3. We study a new 12-fold correspondence relating invariants of Calabi-Yau fourfolds, surfaces and curves. This includes a correspondence between virtual Segre and Verlinde series and improves on the 8-fold correspondence observed in [1].4. We study the higher rank Nekrasov genus and its cohomological limit both of which can be expressed in terms of the Mac-Mahon series M (q) = n>0 (1 − q n ) −n .Along the way, we proved a new combinatorial identity related to Lagrange inversion which appeared in the companion paper [6].