arXiv subjects

Emma Yu Jin

Publications and source records attributed to Emma Yu Jin.

At least 19 recordsLinked to original sources

arXiv2026

Hall-Littlewood functions in noncommuting variables

In 2022 Aliniaeifard, Li, and van Willigenburg defined Schur functions in the algebra of symmetric functions in noncommuting variables (NCSym), answering an open question posed by Rosas and Sagan in 2004. These Schur functions are not monomial positive, since they are defined via a noncommutative analogue of the Jacobi-Trudi determinant. We introduce Hall-Littlewood functions ${\bf P}_{\pi}({\bf x};t)$ indexed by set partitions $\pi$ in noncommuting variables ${\bf x}=({\bf x}_1,{\bf x}_2,\ldots)$, and define Schur functions in noncommuting variables to be ${\bf s}_{\pi}({\bf x})={\bf P}_{\pi}({\bf x};0)$. We prove that the set of Hall-Littlewood functions $\{{\bf P}_{\pi}({\bf x};t)\}$ for all set partitions $\pi$ of $[n]$ forms a $\mathbb{Q}[t]$-basis of NCSym of homogeneous degree $n$, and that this basis is invariant under any permutation acting on set partitions. These Hall-Littlewood functions in NCSym map to classical Hall-Littlewood functions under commutation, up to a scalar factor. We also show that the Hall-Littlewood functions ${\bf P}_{\pi}({\bf x};t)$ naturally refine the lifted Hall-Littlewood functions in NCSym. Specifically, the Schur functions ${\bf s}_{\pi}({\bf x})$ are monomial positive and refine the lifted Schur function introduced by Rosas and Sagan. Moreover, we introduce a star product of two polynomials in NCSym and develop the star-multiplication rule for a lifted and a non-lifted Hall-Littlewood functions in NCSym. This rule is a noncommutative analogue of the product rule for two Hall-Littlewood functions and, in particular, of the Littlewood-Richardson rule. Finally, our approach extends to the algebra of quasisymmetric functions in noncommuting variables (NCQSym) indexed by set compositions.

math.CO↗

arXiv2026

Combinatorial formulas for Macdonald polynomials by superizations

In this paper, we derive new combinatorial formulas for symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ and non-symmetric Macdonald polynomials $E_{\gamma}(X;q,t)$, in terms of several new statistics and the major index, for a partition $\lambda$ and a weak composition $\gamma$. Compared to previous formulas, these new formulas contain the fewest terms and lead to explicit $(q,t)$-formulas for the coefficients in the monomial expansion of $P_{\lambda}(X;q,t)$. In particular, the combinatorial formula for $E_{\gamma}(X;q,t)$ extends the one for $E_{\lambda}(X;q,t)$ indexed by a partition $\lambda$, due to Corteel, Mandelshtam and Williams (2022). Three existing formulas for $P_{\lambda}(X;q,t)$ established by Corteel, Mandelshtam and Williams (2022), by Corteel, Haglund, Mandelshtam, Mason and Williams (2022), and by Mandelshtam (2025) are recovered. Our proof relies on two new statistics on super fillings, employing the superization formulas of Haglund--Haiman--Loehr (2005) and Ayyer--Mandelshtam--Martin (2023), together with our recent approach to modified Macdonald polynomials.

math.CO↗

arXiv2025

The monomial expansions of modified Macdonald polynomials

We discover a family $A$ of sixteen statistics on fillings of any given Young diagram and prove new combinatorial formulas for modified Macdonald polynomials, that is, $$\tilde{H}_{\lambda}(X;q,t)=\sum_{\sigma\in T(\lambda)}x^{\sigma}q^{maj(\sigma)}t^{\eta(\sigma)}$$ for each statistic $\eta\in A$. Building upon this new formula, we establish four compact formulas for the modified Macdonald polynomials, namely, $$\tilde{H}_{\lambda}(X;q,t)=\sum_{\sigma}d_{\varepsilon}(\sigma)x^{\sigma}q^{maj(\sigma)}t^{\eta(\sigma)}$$ which is summed over all canonical or dual canonical fillings of a Young diagram and $d_{\varepsilon}(\sigma)$ is a product of $t$-multinomials. Finally, the compact formulas enable us to derive four explicit expressions for the monomial expansion of modified Macdonald polynomials, one of which coincides with the formula given by Garbali and Wheeler (2020).

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arXiv2024

Modified Macdonald polynomials and mu-Mahonian statistics

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 Martin's multiline-queue formula for the stationary distribution of multitype asymmetric simple exclusion processes, Corteel, Haglund, Mandelshtam, Mason and Williams recently introduced the queue inversion statistic $quinv$ and conjectured that the tableaux formula for $\tilde{H}_{\mu}(X;q,t)$ is invariant if the inversion statistic $inv$ is replaced by $quinv$. This was subsequently resolved by Ayyer, Mandelshtam and Martin, who proposed a stronger conjecture on the equivalence of the two refined formulas for $\tilde{H}_{\mu}(X;q,t)$. Our main result confirms this Ayyer--Mandelshtam--Martin conjecture. We establish an equidistribution between the pairs $(inv,maj)$ and $(quinv,maj)$ of $\mu$-Mahonian statistics on any row-equivalency class $[\tau]$, where $\tau$ is a filling of the Young diagram of $\mu$. As a byproduct of our approach, we show that if $\tau$ is a rectangular filling, the triples $(inv,quinv,maj)$ and $(quinv,inv,maj)$ have the same distribution over $[\tau]$.

math.CO↗

arXiv2024

On a Pieri-like rule for the Petrie symmetric functions

A $k$-ribbon tiling is a decomposition of a connected skew diagram into disjoint ribbons of size $k$. In this paper, we establish a connection between a subset of $k$-ribbon tilings and Petrie symmetric functions, thus providing a combinatorial interpretation for the coefficients in a Pieri-like rule for the Petrie symmetric functions due to Grinberg (Algebr. Comb. 5 (2022), no. 5, 947-1013). This also extends a result by Cheng, Chou and Eu et al. (Proc. Amer. Math. Soc. 151 (2023), no. 5, 1839-1854). As a bonus, our findings can be effectively utilized to derive certain specializations.

math.CO↗

arXiv2024

Equality of skew Schur functions in noncommuting variables

The question of classifying when two skew Schur functions are equal is a substantial open problem, which remains unsolved for over a century. In 2022, Aliniaeifard, Li and van Willigenburg introduced skew Schur functions in noncommuting variables, $s_{(\delta,D)}$, where $D$ is a connected skew diagram with $n$ boxes and $\delta$ is a permutation in the symmetric group $S_n$. In this paper, we combine these two and classify when two skew Schur functions in noncommuting variables are equal: $s_{(\delta,D)} = s_{(\tau,T)}$ such that $D\ne T$ if and only if $D$ is a nonsymmetric ribbon, $T$ is the antipodal rotation of $D$ and $\overline{\tau^{-1}\delta}$ is an explicit bijection between two set partitions determined by $D$.

math.CO↗

arXiv2024

Towards equivalent thickened ribbon Schur functions

Two skew diagrams are defined to be equivalent if their corresponding skew Schur functions are equal. The equivalence classes of ribbons (edgewise connected skew diagrams without a $2\times 2$ block of boxes) have been classified by Billera, Thomas and van Willigenburg in 2006. In this paper, we provide a complete characterization of the equivalence classes of connected skew diagrams with exactly one inclusion-maximal $2\times m$ or $m\times 2$ block of boxes for every $m\ge 2$. In particular, the possible sizes of such equivalence classes are one, two, or four, demonstrating that a single $2\times m$ or $m\times 2$ block dramatically reduces the sizes of equivalence classes. Our result confirms special cases of the elusive conjecture on equivalent connected skew diagrams proposed by McNamara and van Willigenburg in 2009.

math.CO↗

arXiv2022

Symmetric generating functions and Euler-Stirling statistics on permutations

We present (bi-)symmetric generating functions for the joint distributions of Euler-Stirling statistics on permutations, including the number of descents ($\mathsf{des}$), inverse descents ($\mathsf{ides}$), the number of left-to-right maxima ($\mathsf{lmax}$), the number of right-to-left maxima ($\mathsf{rmax}$) and the number of left-to-right minima ($\mathsf{lmin}$). We also show how they recover the classical symmetric generating function of permutations due to Carlitz, Roselle and Scoville (1966). Our proofs exploit three different recursive constructions of inversion sequences, bijections on the multiple equidistributions of Euler-Stirling statistics over permutations and transformation formulas of basic hypergeometric series. Furthermore, we establish a new quadruple equidistribution of Euler-Stirling statistics over inversion sequences, as progress towards a conjecture proposed by Schlosser and the author (2020).

math.CO↗

arXiv2020

Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow

We establish the asymptotic normality of the dimension of large-size random Fishburn matrices by a complex-analytic approach. The corresponding dual problem of size distribution under large dimension is also addressed and follows a quadratic type normal limit law. These results represent the first of their kind and solve two open questions raised in the combinatorial literature. They are presented in a general framework where the entries of the Fishburn matrices are not limited to binary or nonnegative integers. The analytic saddle-point approach we apply, based on a powerful transformation for $q$-series due to Andrews and Jelínek, is also useful in solving a conjecture of Stoimenow in Vassiliev invariants.

math.CO↗

arXiv2020

Asymptotics and statistics on Fishburn matrices and their generalizations

A direct saddle-point analysis (without relying on any modular forms, identities or functional equations) is developed to establish the asymptotics of Fishburn matrices and a large number of other variants with a similar sum of-finite-product form for their (formal) general functions. In addition to solving some conjectures, the application of our saddle-point approach to the distributional aspects of statistics on Fishburn matrices is also examined with many new limit theorems characterized, representing the first of their kind for such structures.

math.CO↗

arXiv2020

Proof of a bi-symmetric septuple equidistribution on ascent sequences

It is well known since the seminal work by Bousquet-Mélou, 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 $({\bf2+2})$-free posets and permutations that avoid a bivincular pattern. Different multiply-refined enumerations of ascent sequences and other bijectively equivalent structures have subsequently been extensively studied by various authors. In this paper, our main contributions are 1. a bijective proof of a bi-symmetric septuple equidistribution of statistics on ascent sequences, involving the number of ascents (asc), the number of repeated entries (rep), the number of zeros (zero), the number of maximal entries (max), the number of right-to-left minima (rmin) and two auxiliary statistics; 2. a new transformation formula for non-terminating basic hypergeometric $_4ϕ_3$ series expanded as an analytic function in base $q$ around $q=1$, which is utilized to prove two (bi)-symmetric quadruple equidistributions on ascent sequences. A by-product of our findings includes the affirmation of a conjecture about the bi-symmetric equidistribution between the quadruples of Euler--Stirling statistics (asc,rep,zero,max) and (rep,asc,max,zero) on ascent sequences, that was motivated by a double Eulerian equidistribution due to Foata (1977) and recently proposed by Fu, Lin, Yan, Zhou and the first author (2018).

math.CO↗

arXiv2019

A new decomposition of ascent sequences and Euler--Stirling statistics

As shown by Bousquet-Mélou--Claesson--Dukes--Kitaev (2010), ascent sequences can be used to encode $({\bf2+2})$-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power series $$\sum_{m=1}^{\infty}\prod_{i=1}^m (1-(1-t)^i).$$ In this paper, we present a novel way to recursively decompose ascent sequences, which leads to: (i) a calculation of the Euler--Stirling distribution on ascent sequences, including the numbers of ascents ($\asc$), repeated entries $(\rep)$, zeros ($\zero$) and maximal entries ($\max$). In particular, this confirms and extends Dukes and Parviainen's conjecture on the equidistribution of $\zero$ and $\max$. (ii) a far-reaching generalization of the generating function formula for $(\asc,\zero)$ due to Jelínek. This is accomplished via a bijective proof of the quadruple equidistribution of $(\asc,\rep,\zero,\max)$ and $(\rep,\asc,\rmin,\zero)$, where $\rmin$ denotes the right-to-left minima statistic of ascent sequences. (iii) an extension of a conjecture posed by Levande, which asserts that the pair $(\asc,\zero)$ on ascent sequences has the same distribution as the pair $(\rep,\max)$ on $({\bf2-1})$-avoiding inversion sequences. This is achieved via a decomposition of $({\bf2-1})$-avoiding inversion sequences parallel to that of ascent sequences. This work is motivated by a double Eulerian equidistribution of Foata (1977) and a tempting bi-symmetry conjecture, which asserts that the quadruples $(\asc,\rep,\zero,\max)$ and $(\rep,\asc,\max,\zero)$ are equidistributed on ascent sequences.

math.CO↗

arXiv2018

Graph limits of random unlabelled $k$-trees

We study random unlabelled $k$-dimensional trees by combining the colouring approach by Gainer-Dewar and Gessel (2014) with the cycle pointing method by Bodirsky, Fusy, Kang and Vigerske (2011). Our main applications are Gromov-Hausdorff-Prokhorov and Benjamini-Schramm limits, that describe their asymptotic geometric shape on a global and local scale as the number of hedra tends to infinity.

math.CO↗

arXiv2018

On the shape of random Pólya structures

Panagiotou and Stufler recently proved an important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size $n$ consists of a conditioned critical Galton-Watson tree $C_n$ and many small forests, where with probability tending to one, as $n$ tends to infinity, any forest $F_n(v)$, that is attached to a node $v$ in $C_n$, is maximally of size $\vert F_n(v)\vert=O(\log n)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities. In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements for $\vert F_n(v)\vert$, namely $\vert F_n(v)\vert=Θ(\log n)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given Pólya tree. Third, we derive the limit probability that for a random node $v$ the attached forest $F_n(v)$ is of a given size. Moreover, structural properties of those forests like the number of their components are studied. Finally, we extend all results to other Pólya structures.

math.CO↗

arXiv2016

A note on the scaling limits of random Pólya trees

Panagiotou and Stufler (arXiv:1502.07180v2) recently proved one important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size $n$ consists of a conditioned critical Galton-Watson tree $C_n$ and many small forests, where with probability tending to one as $n$ tends to infinity, any forest $F_n(v)$, that is attached to a node $v$ in $C_n$, is maximally of size $\vert F_n(v)\vert=O(\log n)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities. In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements on the bound of $\vert F_n(v)\vert$, namely $\vert F_n(v)\vert=Θ(\log n)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given Pólya tree. Finally, we derive the limit probability that for a random node $v$ the attached forest $F_n(v)$ is of a given size.

math.CO↗

arXiv2016

Outside nested decompositions of skew diagrams and Schur function determinants

In this paper we describe the thickened strips and the outside nested decompositions of any skew shape $λ/μ$. For any such decomposition $Φ=(Θ_1,Θ_2,\ldots,Θ_g)$ of the skew shape $λ/μ$ where $Θ_i$ is a thickened strip for every $i$, if $r$ is the number of boxes that are contained in any two distinct thickened strips of $Φ$, we establish a determinantal formula of the function $s_{λ/μ}(X)p_{1^r}(X)$ with the Schur functions of thickened strips as entries, where $s_{λ/μ}(X)$ is the Schur function of the skew shape $λ/μ$ and $p_{1^r}(X)$ is the power sum symmetric function index by the partition $(1^r)$. This generalizes Hamel and Goulden's theorem on the outside decompositions of the skew shape $λ/μ$. As an application of our theorem, we derive the number of $m$-strip tableaux which was first counted by Baryshnikov and Romik via extending the transfer operator approach due to Elkies.

math.CO↗

arXiv2016

Graph limits of random graphs from a subset of connected $k$-trees

For any set $Ω$ of non-negative integers such that $\{0,1\}\subseteq Ω$ and $\{0,1\}\ne Ω$, we consider a random $Ω$-$k$-tree ${\sf G}_{n,k}$ that is uniformly selected from all connected $k$-trees of $(n+k)$ vertices where the number of $(k+1)$-cliques that contain any fixed $k$-clique belongs to $Ω$. We prove that ${\sf G}_{n,k}$, scaled by $(kH_{k}σ_Ω)/(2\sqrt{n})$ where $H_{k}$ is the $k$-th Harmonic number and $σ_Ω>0$, converges to the Continuum Random Tree $\mathcal{T}_{\sf e}$. Furthermore, we prove the local convergence of the rooted random $Ω$-$k$-tree ${\sf G}_{n,k}^{\circ}$ to an infinite but locally finite random $Ω$-$k$-tree ${\sf G}_{\infty,k}$.

math.PR↗

arXiv2015

Heaps and Two Exponential Structures

Take ${\sf Q}=({\sf Q}_1,{\sf Q}_2,\ldots)$ to be an exponential structure and $M(n)$ to be the number of minimal elements of ${\sf Q}_n$ where $M(0)=1$. Then a sequence of numbers $\{r_n({\sf Q}_n)\}_{n\ge 1}$ is defined by the equation \begin{eqnarray*} \sum_{n\ge 1}r_n({\sf Q}_n)\frac{z^n}{n!\,M(n)}=-\log(\sum_{n\ge 0}(-1)^n\frac{z^n}{n!\,M(n)}). \end{eqnarray*} Let $\bar{\sf Q}_n$ denote the poset ${\sf Q}_n$ with a $\hat{0}$ adjoined and let $\hat{1}$ denote the unique maximal element in the poset ${\sf Q}_n$. Furthermore, let $μ_{{\sf Q}_n}$ be the Möbius function on the poset $\bar{\sf Q}_n$. Stanley proved that $r_n({\sf Q}_n)=(-1)^nμ_{{\sf Q}_n}(\hat{0},\hat{1})$. This implies that the numbers $r_n({\sf Q}_n)$ are integers. In this paper, we study the cases ${\sf Q}_n=Π_n^{(r)}$ and ${\sf Q}_n={\sf Q}_n^{(r)}$ where $Π_n^{(r)}$ and ${\sf Q}_n^{(r)}$ are posets, respectively, of set partitions of $[rn]$ whose block sizes are divisible by $r$ and of $r$-partitions of $[n]$. In both cases we prove that $r_n(Π_n^{(r)})$ and $r_n({\sf Q}_n^{(r)})$ enumerate the pyramids by applying the Cartier-Foata monoid identity and further prove that $r_n(Π_n^{(r)})$ is the generalized Euler number $E_{rn-1}$ and that $r_n({\sf Q}_n^{(2)})$ is the number of complete non-ambiguous trees of size $2n-1$ by bijections. This gives a new proof of Welker's theorem that $r_n(Π_n^{(r)})=E_{rn-1}$ and implies the construction of $r$-dimensional complete non-ambiguous trees. As a bonus of applying the theory of heaps, we establish a bijection between the set of complete non-ambiguous forests and the set of pairs of permutations with no common rise. This answers an open question raised by Aval {\it et al.}.

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