Related papers: Proof of Han's Hook Expansion Conjecture
Let $b_{t,i}(n)$ denote the total number of $i$-hooks in $t$-regular partitions of $n$. Singh and Barman conjectured that $b_{t+1,2}(n) \geq b_{t,2}(n)$ holds for all $t\ge 3$ and $n\ge 0$. This conjecture was known to hold for $t=3$ due to…
In this paper, we give a refinement of a theorem by Franks, which answers two questions raised by Kang.
In this paper, we present a direct bijective proof of the hook-length formula for standard immaculate tableaux, which arose in the study of non-commutative symmetric functions. Our proof is along the spirit of Novelli, Pak and…
We provide a proof of the Borwein Conjecture using analytic methods.
Two conjectures recently proposed by one of the authors are disproved
We provide a combinatorial proof of an infinite extension of the Hales--Jewett theorem due to T. Carlson and independently due to H. Furstenberg and Y. Katznelson
Coinduction occurs in two guises in Horn clause logic: in proofs of self-referencing properties and relations, and in proofs involving construction of (possibly irregular) infinite data. Both instances of coinductive reasoning appeared in…
The original motivation for study for hook length polynomials was to find a combinatorial proof for a hook length formula for binary trees given by Postnikov, as well as a proof for a hook length polynomial formula conjectured by Lascoux.…
We propose a proof of the Lagrange Interpolation Formula based on the Chinese Remainder Theorem for arbitrary rings. Even such relationships are known, we think that our viewpoint is worth being published.
We present an improved incremental selection algorithm of the selection algorithm presented in [1] and prove all the selected conjectures.
We survey recent joint work with M. Rapoport and W. Zhang related to the arithmetic Gan-Gross-Prasad conjecture for Shimura varieties attached to unitary groups.
We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path…
Using a relative trace formula approach, we prove the twisted global Gan-Gross-Prasad conjecture for $\operatorname{U}(V) \subseteq \operatorname{GL}(V)$, as well as its refinement, under some unramifiedness assumptions and local conditions…
We establish an expansion theory for $\text{SL}_2(\mathbb Z/q\mathbb Z)$. Incorporating this into a framework recently developed by Shkredov, we confirm Zaremba's conjecture.
In this article, we prove a weighted version of Saitoh's conjecture. As an application, we prove a weighted version of Saitoh's conjecture for higher derivatives.
We prove the volume conjecture for any twist knots by using an equivalence relation, complex analysis, analytic continuation, and function of several complex variables on the basis of colored Jones polynomials.
We find two new hook length formulas for binary trees. The particularity of our formulas is that the hook length $h_v$ appears as an exponent.
The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement…
We give a simple proof of the $A_2$ conjecture proved recently by T. Hyt\"onen. Our proof avoids completely the notion of the Haar shift operator, and it is based only on the "local mean oscillation decomposition". Also our proof yields a…
We prove a generalization of Maehara's lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig's interpolation property. As a…