Jianxi Mao

• PhD
• PhD Student at Dalian University of Technology

Let n and b be positive integers. Define the amazing matrix Pn,b=[P(i,j)]i,j=0n−1 to be an n×n matrix with entriesP(i,j)=1bn∑r≥0(−1)r(n+1r)(n−1−i+(j+1−r)bn). Diaconis and Fulman conjectured that the amazing matrix is totally positive. We give an affirmative answer to this conjecture.

Let R=R(d(t),h(t)) be a Riordan array, where d(t)=∑n≥0dntn and h(t)=∑n≥0hntn. We show that if the matrix[d0h000⋯d1h1h00d2h2h1h0⋮⋮⋱] is totally positive, then so is the Riordan array R.

Let $R=\mathcal{R}(d(t),h(t))$ be a Riordan array, where $d(t)=\sum_{n\ge 0}d_nt^n$ and $h(t)=\sum_{n\ge 0}h_nt^n$. We show that if the matrix \begin{equation*} \left[\begin{array}{ccccc} d_0 & h_0 & 0 & 0 &\cdots\\ d_1 & h_1 & h_0 & 0 &\\ d_2 & h_2 & h_1 & h_0 &\\ \vdots&\vdots&&&\ddots \end{array}\right] \end{equation*} is totally positive, then...

We construct bijections to show that two pairs of multiple set-valued statistics of permutations are equidistributed on symmetric groups. This extends a recent result of Sokal and the second author valid for integer-valued statistics as well as a previous result of Foata and Han for bivariable set-valued statistics.

In a recent paper ({arXiv:2101.01928v1}) Baril and Kirgizov posed two conjectures on the equidistibution of $(cyc, des_2)\sim (cyc, pex)$ and $(des, des_2)\sim (exc, pex)$, where cyc, des and exc are classical statistics counting the numbers of cycles, descents and excedances of permutation, while $des_2$ and $pex$ are numbers of special descents a...

In this paper, we show that the numbers of t -stack sortable n -permutations with k − 1 descents satisfy central and local limit theorems for t = 1, 2, n − 1 and n − 2. This result, in particular, gives an affirmative answer to Shapiro's question about the asymptotic normality of the Narayana numbers.

A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridge's formula by using cycle peaks and excedances of permutations. We prove a series of new general formulae expressing polynomials counting p...

It is well known that the permutation peak polynomials and descent polynomials are connected via a quadratic transformation. By rephrasing the latter formula with permutation cycle peaks and excedances we are able to prove a series of general formulas expressing polynomials counting permutations by various excedance statistics in terms of refined E...

We construct bijections to show that two pairs of sextuple set-valued statistics of permutations are equidistributed on symmetric groups. This extends a recent result of Sokal and the second author valid for integer-valued statistics as well as a previous result of Foata and Han for bivariable set-valued statistics.

A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. Rephrasing the latter formula with permutation cycle peaks and excedances we are able to prove a series of general formulas expressing polynomials counting permutations by various excedance statistics in terms...

Let R=[rn,k]n,k≥0 be a Riordan array. Define the row polynomials Rn(q)=∑k=0ⁿrn,kqk and the row polynomial matrix R(q)=[rn,k(q)]n,k≥0 by rn,k(q)=∑j=kⁿrn,jqj−k. Then R(q) is also a Riordan array with the Rn(q) located on the leftmost column of R(q). In this paper we investigate combinatorial properties of the matrix R(q) and the sequence (Rn(q))n≥0,...