相关论文: A maj-inv bijection for C_2 \wr A_n
We came across an unexpected connection between a remarkable grammar of Dumont for the joint distribution of $(\exc, \fix)$ over $S_n$ and a beautiful theorem of Diaconis-Evans-Graham on successions and fixed points of permutations. With…
As natural generalizations of the descent number ($\des$) and the major index ($\maj$), Rawlings introduced the notions of the $r$-descent number ($r\des$) and the $r$-major index ($r\maj$) for a given positive integer $r$. A pair $(\st_1,…
It is well known since the seminal work by Bousquet-M\'elou, Claesson, Dukes and Kitaev (2010) that certain refinements of the ascent sequences with respect to several natural statistics are in bijection with corresponding refinements of…
Prime counting functions are believed to exhibit, in various contexts, discrepancies beyond what famous equidistribution results predict; this phenomenon is known as Chebyshev's bias. Rubinstein and Sarnak have developed a framework which…
Let B_n be the hyperoctahedral group; that is, the set of all signed permutations on n letters, and let B_n(T) be the set of all signed permutations in B_n which avoids a set T of signed patterns. In this paper, we find all the…
The Haglund--Haiman--Loehr theorem provides the following combinatorial formula for the modified Macdonald polynomials: $$\tilde{H}_{\mu}(X;q,t)=\sum_{\sigma: \mu\rightarrow \mathbb{P}}x^{\sigma}t^{maj(\sigma)}q^{inv(\sigma)}.$$ Inspired by…
We prove an identity [Eq. (1) below] among SU(2) 6j and 9j symbols that generalizes the Biedenharn-Elliott sum rule. We prove the result using diagrammatic techniques (briefly reviewed here), and then provide an algebraic proof. This…
We provide a bijective map from the partitions enumerated by the series side of the Rogers-Selberg mod 7 identities onto partitions associated with a special case of Basil Gordon's combinatorial generalization of the Rogers-Ramanujan…
We give two simple proofs of a conjecture of Richard Stanley concerning the equidistribution of derangements and alternating permutations with the maximal number of fixed points.
This paper investigates two involutions on binary trees. One is the mirror symmetry of binary trees which combined with the classical bijection $\varphi$ between binary trees and plane trees answers an open problem posed by Bai and Chen.…
Given a real reductive group Lie group $G_\mathbb{R}$, the Mackey analogy is a bijection between the set of irreducible tempered representations of $G_\mathbb{R}$ and the set of irreducible unitary representations of its Cartan motion…
Permutation tableaux are combinatorial objects related with permutations and various statistics on them. They appeared in connection with total positivity in Grassmannians, and stationary probabilities in a PASEP model. In particular they…
We examine an identity originally stated in Ramanujan's ``lost notebook'' and first proven algebraically by Andrews and combinatorially by Kim. We give two independent combinatorial proofs and interpretations of this identity, which also…
Fix a monoidal category C. The 2-category of monads in the 2-category of C-actegories, colax C-equivarant functors, and C-equivariant natural transformations of colax functors, may be recast in terms of pairs consisting of a usual monad and…
A new q-binomial theorem for Macdonald polynomials is employed to prove an A_n analogue of the celebrated Selberg integral. This confirms the g=A_n case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral…
We extend the family of statistics maj_d, introduced for permutations by Kadell (1985), to standard Young tableaux. At one extreme, we have the traditional Major index statistic maj_1 for tableaux. At the other end, maj_n = inv, the…
Majorization inequalities have a long history, going back to Maclaurin and Newton. They were recently studied for several families of symmetric functions, including by Cuttler--Greene--Skandera (2011), Sra (2016), Khare--Tao (2021),…
In [16] the fundamental relationship between stable quotient invariants and the B-model for local P2 in all genera was studied under some specialization of equivariant variables. We generalize the argument of [16] to full equivariant…
Guibert and Linusson introduced the family of doubly alternating Baxter permutations, i.e. Baxter permutations $\sigma \in S_n$, such that $\sigma$ and $\sigma^{-1}$ are alternating. They proved that the number of such permutations in…
We prove that every partially ordered set on $n$ elements contains $k$ subsets $A_{1},A_{2},\dots,A_{k}$ such that either each of these subsets has size $\Omega(n/k^{5})$ and, for every $i<j$, every element in $A_{i}$ is less than or equal…