Lattice Points in the Newton Polytopes of Key Polynomials
We confirm a conjecture of Monical, Tokcan and Yong on a characterization of the lattice points in the Newton polytopes of key polynomials.
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arXiv subjects
Neil J. Y. Fan
Publications and source records attributed to Neil J. Y. Fan.
Set-valued Rothe Tableaux and Grothendieck Polynomials
The notion of set-valued Young tableaux was introduced by Buch in his study of the Littlewood-Richardson rule for stable Grothendieck polynomials. Knutson, Miller and Yong showed that the double Grothendieck polynomials of 2143-avoiding permutations can be generated by set-valued Young tableaux. In this paper, we introduce the structure of set-valued Rothe tableaux of permutations. Given the Rothe diagram $D(w)$ of a permutation $w$, a set-valued Rothe tableau of shape $D(w)$ is a filling of finite nonempty subsets of positive integers into the squares of $D(w)$ such that the rows are weakly decreasing and the columns are strictly increasing. We show that the double Grothendieck polynomials of 1432-avoiding permutations can be generated by set-valued Rothe tableaux. When restricted to 321-avoiding permutations, our formula specializes to the tableau formula for double Grothendieck polynomials due to Matsumura. Employing the properties of tableau complexes given by Knutson, Miller and Yong, we obtain two alternative tableau formulas for the double Grothendieck polynomials of 1432-avoiding permutations.
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Bumpless Pipedreams, Reduced Word Tableaux and Stanley Symmetric Functions
Lam, Lee and Shimozono introduced the structure of bumpless pipedreams in their study of back stable Schubert calculus. They found that a specific family of bumpless pipedreams, called EG-pipedreams, can be used to interpret the Edelman-Greene coefficients appearing in the expansion of a Stanley symmetric function in the basis of Schur functions. It is well known that the Edelman-Greene coefficients can also be interpreted in terms of reduced word tableaux for permutations. Lam, Lee and Shimozono proposed the problem of finding a shape preserving bijection between reduced word tableaux for a permutation $w$ and EG-pipedreams of $w$. In this paper, we construct such a bijection. The key ingredients are two new developed isomorphic tree structures associated to $w$: the modified Lascoux-Schützenberger tree of $w$ and the Edelman-Greene tree of $w$. Using the Little map, we show that the leaves in the modified Lascoux-Schützenberger of $w$ are in bijection with the reduced word tableaux for $w$. On the other hand, applying the droop operation on bumpless pipedreams also introduced by Lam, Lee and Shimozono, we show that the leaves in the Edelman-Greene tree of $w$ are in bijection with the EG-pipedreams of $w$. This allows us to establish a shape preserving one-to-one correspondence between reduced word tableaux for $w$ and EG-pipedreams of $w$.
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Proof of a Conjecture of Reiner-Tenner-Yong on Barely Set-valued Tableaux
The notion of a barely set-valued semistandard Young tableau was introduced by Reiner, Tenner and Yong in their study of the probability distribution of edges in the Young lattice of partitions. Given a partition $λ$ and a positive integer $k$, let ${\mathrm{BSSYT}}(λ,k)$ (respectively, ${\mathrm{SYT}}(λ,k)$) denote the set of barely set-valued semistandard Young tableaux (respectively, ordinary semistandard Young tableaux) of shape $λ$ with entries in row $i$ not exceeding $k+i$. In the case when $λ$ is a rectangular staircase partition $δ_d(b^a)$, Reiner, Tenner and Yong conjectured that $|{\mathrm{BSSYT}}(λ,k)|= \frac{kab(d-1)}{(a+b)} |{\mathrm{SYT}}(λ,k)|$. In this paper, we establish a connection between barely set-valued tableaux and reverse plane partitions with designated corners. We show that for any shape $λ$, the expected jaggedness of a subshape of $λ$ under the weak probability distribution can be expressed as $\frac{2|{\mathrm{BSSYT}}(λ,k)|} {k|{\mathrm{SYT}}(λ,k)|}$. On the other hand, when $λ$ is a balanced shape with $r$ rows and $c$ columns, Chan, Haddadan, Hopkins and Moci proved that the expected jaggedness of a subshape in $λ$ under the weak distribution equals $2rc/(r+c)$. Hence, for a balanced shape $λ$ with $r$ rows and $c$ columns, we establish the relation that $|{\mathrm{BSSYT}}(λ,k)|=\frac{krc}{(r+c)}|{\mathrm{SYT}}(λ,k)|$. Since a rectangular staircase shape $δ_d(b^a)$ is a balanced shape, we confirm the conjecture of Reiner, Tenner and Yong.
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Standard Rothe Tableaux
Edelman and Greene constructed a bijection between the set of standard Young tableaux and the set of balanced Young tableaux of the same shape. Fomin, Greene, Reiner and Shimozono introduced the notion of balanced Rothe tableaux of a permutation $w$, and established a bijection between the set of balanced Rothe tableaux of $w$ and the set of reduced words of $w$. In this paper, we introduce the notion of standard Rothe tableaux of $w$, which are tableaux obtained by labelling the cells of the Rothe diagram of $w$ such that each row and each column is increasing. We show that the number of standard Rothe tableaux of $w$ is smaller than or equal to the number of balanced Rothe tableaux of $w$, with equality if and only if $w$ avoids the four patterns 2413, 2431, 3142 and 4132. When $w$ is a dominant permutation, i.e., 132-avoiding, the Rothe diagram of $w$ is a Young diagram, so this reduces to the result of Edelman and Greene.
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On parabolic Kazhdan-Lusztig R-polynomials for the symmetric group
Parabolic $R$-polynomials were introduced by Deodhar as parabolic analogues of ordinary $R$-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic $R$-polynomials for the symmetric group. Let $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$, and let $S=\{s_i\,|\, 1\leq i\leq n-1\}$ be the generating set of $S_n$, where for $1\leq i\leq n-1$, $s_i$ is the adjacent transposition. For a subset $J\subseteq S$, let $(S_n)_J$ be the parabolic subgroup generated by $J$, and let $(S_n)^{J}$ be the set of minimal coset representatives for $S_n/(S_n)_J$. For $u\leq v\in (S_n)^J$ in the Bruhat order and $x\in \{q,-1\}$, let $R_{u,v}^{J,x}(q)$ denote the parabolic $R$-polynomial indexed by $u$ and $v$. Brenti found a formula for $R_{u,v}^{J,x}(q)$ when $J=S\setminus\{s_i\}$, and obtained an expression for $R_{u,v}^{J,x}(q)$ when $J=S\setminus\{s_{i-1},s_i\}$. We introduce a statistic on pairs of permutations in $(S_n)^J$ for $J=S\setminus\{s_{i-2},s_{i-1},s_i\}$. Then we give a formula for $R_{u,v}^{J,x}(q)$, where $J=S\setminus\{s_{i-2},s_{i-1},s_i\}$ and $i$ appears after $i-1$ in $v$. We also pose a conjecture for $R_{u,v}^{J,x}(q)$, where $J=S\setminus\{s_{k},s_{k+1},\ldots,s_i\}$ with $1\leq k\leq i\leq n-1$ and the elements $k+1,k+2,\ldots, i$ appear in increasing order in $v$.
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A Class of Kazhdan-Lusztig R-Polynomials and q-Fibonacci Numbers
Let $S_n$ denote the symmetric group on $\{1,2,\ldots,n\}$. For two permutations $u, v\in S_n$ such that $u\leq v$ in the Bruhat order, let $R_{u,v}(q)$ and $\R_{u,v}(q)$ denote the Kazhdan-Lusztig $R$-polynomial and $\R$-polynomial, respectively. Let $v_n=34\cdots n\, 12$, and let $σ$ be a permutation such that $σ\leq v_n$. We obtain a formula for the $\R$-polynomials $\R_{σ,v_n}(q)$ in terms of the $q$-Fibonacci numbers depending on a parameter determined by the reduced expression of $σ$. When $σ$ is the identity $e$, this reduces to a formula obtained by Pagliacci. In another direction, we obtain a formula for the $\R$-polynomial $\R_{e,\,v_{n,i}}(q)$, where $v_{n,i} = 3 4\cdots i\,n\, (i+1)\cdots (n-1)\, 12$. In a more general context, we conjecture that for any two permutations $σ,τ\in S_n$ such that $σ\leq τ\leq v_n$, the $\R$-polynomial $\R_{σ,τ}(q)$ can be expressed as a product of $q$-Fibonacci numbers multiplied by a power of $q$.
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Combinatorial Proof of the Inversion Formula on the Kazhdan-Lusztig R-Polynomials
Let $W$ be a Coxeter group, and for $u,v\in W$, let $R_{u,v}(q)$ be the Kazhdan-Lusztig $R$-polynomial indexed by $u$ and $v$. In this paper, we present a combinatorial proof of the inversion formula on $R$-polynomials due to Kazhdan and Lusztig. This problem was raised by Brenti. Based on Dyer's combinatorial interpretation of the $R$-polynomials in terms of increasing Bruhat paths, we reformulate the inversion formula in terms of $V$-paths. By a $V$-path from $u$ to $v$ with bottom $w$ we mean a pair $(Δ_1,Δ_2)$ of Bruhat paths such that $Δ_1$ is a decreasing path from $u$ to $w$ and $Δ_2$ is an increasing path from $w$ to $v$. We find a reflection principle on $V$-paths, which leads to a combinatorial proof of the inversion formula. Moreover, we give two applications of the reflection principle. First, we restrict this involution to $V$-paths from $u$ to $v$ with maximal length. This provides a direct interpretation for the equi-distribution property that any nontrivial interval $[u,v]$ has as many elements of even length as elements of odd length. This property was obtained by Verma in his derivation of the Möbius function of the Bruhat order. Second, using the reflection principle for the symmetric group, we obtain a refinement of the inversion formula by restricting the summation to permutations ending with a given element.
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Partitions and Partial Matchings Avoiding Neighbor Patterns
We obtain the generating functions for partial matchings avoiding neighbor alignments and for partial matchings avoiding neighbor alignments and left nestings. We show that there is a bijection between partial matchings avoiding three neighbor patterns (neighbor alignments, left nestings and right nestings) and set partitions avoiding right nestings via an intermediate structure of integer compositions. Such integer compositions are known to be in one-to-one correspondence with self-modified ascent sequences or $3\bar{1}52\bar{4}$-avoiding permutations, as shown by Bousquet-Mélou, Claesson, Dukes and Kitaev.
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Labeled Ballot Paths and the Springer Numbers
The Springer numbers are defined in connection with the irreducible root systems of type $B_n$, which also arise as the generalized Euler and class numbers introduced by Shanks. Combinatorial interpretations of the Springer numbers have been found by Purtill in terms of Andre signed permutations, and by Arnol'd in terms of snakes of type $B_n$. We introduce the inversion code of a snake of type $B_n$ and establish a bijection between labeled ballot paths of length n and snakes of type $B_n$. Moreover, we obtain the bivariate generating function for the number B(n,k) of labeled ballot paths starting at (0,0) and ending at (n,k). Using our bijection, we find a statistic $α$ such that the number of snakes $π$ of type $B_n$ with $α(π)=k$ equals B(n,k). We also show that our bijection specializes to a bijection between labeled Dyck paths of length 2n and alternating permutations on [2n].
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Han's Bijection via Permutation Codes
We show that Han's bijection when restricted to permutations can be carried out in terms of the cyclic major code and the cyclic inversion code. In other words, it maps a permutation $π$ with a cyclic major code $(s_1, s_2, ..., s_n)$ to a permutation $σ$ with a cyclic inversion code $(s_1,s_2, ..., s_n)$. We also show that the fixed points of Han's map can be characterized by the strong fixed points of Foata's second fundamental transformation. The notion of strong fixed points is related to partial Foata maps introduced by Björner and Wachs.
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The Generating Function for the Dirichlet Series $L_m(s)$
The Dirichlet series $L_m(s)$ are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with these Dirichlet series, denoted by $\{s_{m,n}\}_{n\geq 0}$. We obtain a formula for the exponential generating function $s_m(x)$ of $s_{m,n}$, where m is an arbitrary positive integer. In particular, for m>1, say, $m=bu^2$, where b is square-free and u>1, we prove that $s_m(x)$ can be expressed as a linear combination of the four functions $w(b,t)\sec (btx)(\pm \cos ((b-p)tx)\pm \sin (ptx))$, where p is an integer satisfying $0\leq p\leq b$, $t|u^2$ and $w(b,t)=K_bt/u$ with $K_b$ being a constant depending on b. Moreover, the Dirichlet series $L_m(s)$ can be easily computed from the generating function formula for $s_m(x)$. Finally, we show that the main ingredient in the formula for $s_{m,n}$ has a combinatorial interpretation in terms of the m-signed permutations defined by Ehrenborg and Readdy. In principle, this answers a question posed by Shanks concerning a combinatorial interpretation for the numbers $s_{m,n}$.
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