@article{barkley_liu_2021, title={Channels, Billiards, and Perfect Matching 2-Divisibility}, volume={28}, ISSN={["1077-8926"]}, DOI={10.37236/9151}, abstractNote={Let $m_G$ denote the number of perfect matchings of the graph $G$. We introduce a number of combinatorial tools for determining the parity of $m_G$ and giving a lower bound on the power of 2 dividing $m_G$. In particular, we introduce certain vertex sets called channels, which correspond to elements in the kernel of the adjacency matrix of $G$ modulo $2$. A result of Lovász states that the existence of a nontrivial channel is equivalent to $m_G$ being even. We give a new combinatorial proof of this result and strengthen it by showing that the number of channels gives a lower bound on the power of $2$ dividing $m_G$ when $G$ is planar. We describe a number of local graph operations which preserve the number of channels. We also establish a surprising connection between 2-divisibility of $m_G$ and dynamical systems by showing an equivalency between channels and billiard paths. We exploit this relationship to show that $2^{\frac{\gcd(m+1,n+1)-1}{2}}$ divides the number of domino tilings of the $m\times n$ rectangle. We also use billiard paths to give a fast algorithm for counting channels (and hence determining the parity of the number of domino tilings) in simply connected regions of the square grid.}, number={2}, journal={ELECTRONIC JOURNAL OF COMBINATORICS}, author={Barkley, Grant T. and Liu, Ricky Ini}, year={2021}, month={Jun} }
 @article{hatam_johnson_liu_macaulay_2021, title={Determinantal Formulas for SEM Expansions of Schubert Polynomials}, ISSN={["0219-3094"]}, DOI={10.1007/s00026-021-00558-z}, abstractNote={We show that for any permutation w that avoids a certain set of 13 patterns of length 5 and 6, the Schubert polynomial $${\mathfrak {S}}_w$$ can be expressed as the determinant of a matrix of elementary symmetric polynomials in a manner similar to the Jacobi–Trudi identity. For such w, this determinantal formula is equivalent to a (signed) subtraction-free expansion of $$\mathfrak S_w$$ in the basis of standard elementary monomials.}, journal={ANNALS OF COMBINATORICS}, author={Hatam, Hassan and Johnson, Joseph and Liu, Ricky Ini and Macaulay, Maria}, year={2021}, month={Oct} }
 @article{liu_weselcouch_2021, title={P-Partitions and Quasisymmetric Power Sums}, volume={2021}, ISSN={["1687-0247"]}, DOI={10.1093/imrn/rnz375}, abstractNote={
 The $(P, \omega )$-partition generating function of a labeled poset $(P, \omega )$ is a quasisymmetric function enumerating certain order-preserving maps from $P$ to ${\mathbb{Z}}^+$. We study the expansion of this generating function in the recently introduced type 1 quasisymmetric power sum basis $\{\psi _{\alpha }\}$. Using this expansion, we show that connected, naturally labeled posets have irreducible $P$-partition generating functions. We also show that series-parallel posets are uniquely determined by their partition generating functions. We conclude by giving a combinatorial interpretation for the coefficients of the $\psi _{\alpha }$-expansion of the $(P, \omega )$-partition generating function akin to the Murnaghan–Nakayama rule.}, number={18}, journal={INTERNATIONAL MATHEMATICS RESEARCH NOTICES}, author={Liu, Ricky Ini and Weselcouch, Michael}, year={2021}, month={Sep}, pages={13975–14015} }
 @article{liu_smith_2021, title={Up- and Down-operators on Young's Lattice}, volume={28}, ISSN={["1077-8926"]}, DOI={10.37236/10099}, abstractNote={The up-operators $u_i$ and down-operators $d_i$ (introduced as Schur operators by Fomin) act on partitions by adding/removing a box to/from the $i$th column if possible. It is well known that the $u_i$ alone satisfy the relations of the (local) plactic monoid, and the present authors recently showed that relations of degree at most 4 suffice to describe all relations between the up-operators. Here we characterize the algebra generated by the up- and down-operators together, showing that it can be presented using only quadratic relations.}, number={3}, journal={ELECTRONIC JOURNAL OF COMBINATORICS}, author={Liu, Ricky Ini and Smith, Christian}, year={2021}, month={Jul} }
 @article{liu_weselcouch_2020, title={P-partition generating function equivalence of naturally labeled posets}, volume={170}, ISSN={["1096-0899"]}, DOI={10.1016/j.jcta.2019.105136}, abstractNote={The P-partition generating function of a (naturally labeled) poset P is a quasisymmetric function enumerating order-preserving maps from P to Z+. Using the Hopf algebra of posets, we give necessary conditions for two posets to have the same generating function. In particular, we show that they must have the same number of antichains of each size, as well as the same shape (as defined by Greene). We also discuss which shapes guarantee uniqueness of the P-partition generating function and give a method of constructing pairs of non-isomorphic posets with the same generating function.}, journal={JOURNAL OF COMBINATORIAL THEORY SERIES A}, author={Liu, Ricky Ini and Weselcouch, Michael}, year={2020}, month={Feb} }
 @article{liu_smith_2020, title={The algebra of Schur operators}, volume={87}, ISSN={["1095-9971"]}, DOI={10.1016/j.ejc.2020.103130}, abstractNote={We study a representation of the (local) plactic monoid given by Schur operators ui, which act on partitions by adding a box in column i (if possible). In particular, we give a complete list of the relations that hold in the algebra of Schur operators.}, journal={EUROPEAN JOURNAL OF COMBINATORICS}, author={Liu, Ricky Ini and Smith, Christian}, year={2020}, month={Jun} }
 @article{liu_morales_mészáros_2019, title={Flow Polytopes and the Space of Diagonal Harmonics}, ISSN={0008-414X 1496-4279}, url={http://dx.doi.org/10.4153/CJM-2018-007-3}, DOI={10.4153/CJM-2018-007-3}, abstractNote={Abstract A result of Haglund implies that the $(q,t)$ -bigraded Hilbert series of the space of diagonal harmonics is a $(q,t)$ -Ehrhart function of the flow polytope of a complete graph with netflow vector $(-n,1,\ldots ,1)$ . We study the $(q,t)$ -Ehrhart functions of flow polytopes of threshold graphs with arbitrary netflow vectors. Our results generalize previously known specializations of the mentioned bigraded Hilbert series at $t=1$ , $0$ , and $q^{-1}$ . As a corollary to our results, we obtain a proof of a conjecture of Armstrong, Garsia, Haglund, Rhoades, and Sagan about the $(q,q^{-1})$ -Ehrhart function of the flow polytope of a complete graph with an arbitrary netflow vector.}, journal={Canadian Journal of Mathematics}, publisher={Canadian Mathematical Society}, author={Liu, Ricky Ini and Morales, Alejandro H. and Mészáros, Karola}, year={2019}, month={Jan}, pages={1–27} }
 @article{liu_meszaros_st dizier_2019, title={GELFAND-TSETLIN POLYTOPES: A STORY OF FLOW AND ORDER POLYTOPES}, volume={33}, ISBN={1095-7146}, DOI={10.1137/19M1251242}, abstractNote={Gelfand-Tsetlin polytopes are prominent objects in algebraic combinatorics. The number of integer points of the Gelfand-Tsetlin polytope $\mathrm{GT}(\lambda)$ is equal to the dimension of the corresponding irreducible representation of $GL(n)$. It is well-known that the Gelfand-Tsetlin polytope is a marked order polytope; the authors have recently shown it to be a flow polytope. In this paper, we draw corollaries from this result and establish a general theory connecting marked order polytopes and flow polytopes.}, number={4}, journal={SIAM JOURNAL ON DISCRETE MATHEMATICS}, author={Liu, Ricky I and Meszaros, Karola and St Dizier, Avery}, year={2019}, pages={2394–2415} }
 @article{blasiak_liu_2018, title={Kronecker coefficients and noncommutative super Schur functions}, volume={158}, ISSN={0097-3165}, url={http://dx.doi.org/10.1016/j.jcta.2018.02.007}, DOI={10.1016/j.jcta.2018.02.007}, abstractNote={Abstract The theory of noncommutative Schur functions can be used to obtain positive combinatorial formulae for the Schur expansion of various classes of symmetric functions, as shown by Fomin and Greene [11] . We develop a theory of noncommutative super Schur functions and use it to prove a positive combinatorial rule for the Kronecker coefficients g λ μ ν where one of the partitions is a hook, recovering previous results of the two authors [7] , [22] . This method also gives a precise connection between this rule and a heuristic for Kronecker coefficients first investigated by Lascoux [19] .}, journal={Journal of Combinatorial Theory, Series A}, publisher={Elsevier BV}, author={Blasiak, Jonah and Liu, Ricky Ini}, year={2018}, month={Aug}, pages={315–361} }
 @article{liu_2017, title={A simplified Kronecker rule for one hook shape}, volume={145}, ISSN={0002-9939 1088-6826}, url={http://dx.doi.org/10.1090/proc/13692}, DOI={10.1090/proc/13692}, abstractNote={Recently Blasiak gave a combinatorial rule for the Kronecker coefficient $g_{\lambda \mu \nu}$ when $\mu$ is a hook shape by defining a set of colored Yamanouchi tableaux with cardinality $g_{\lambda\mu\nu}$ in terms of a process called conversion. We give a characterization of colored Yamanouchi tableaux that does not rely on conversion, which leads to a simpler formulation and proof of the Kronecker rule for one hook shape.}, number={9}, journal={Proceedings of the American Mathematical Society}, publisher={American Mathematical Society (AMS)}, author={Liu, Ricky Ini}, year={2017}, month={May}, pages={3657–3664} }
 @article{liu_2016, title={Complete branching rules for Specht modules}, volume={446}, ISSN={["1090-266X"]}, DOI={10.1016/j.jalgebra.2015.09.008}, abstractNote={We give a combinatorial description for when the Specht module of an arbitrary diagram admits a (complete) branching rule. This description, given in terms of the maximal rectangles of the diagram, generalizes all previously known branching rules for Specht modules, such as those given by Reiner and Shimozono for northwest diagrams and by the present author for forest diagrams.}, journal={JOURNAL OF ALGEBRA}, author={Liu, Ricky Ini}, year={2016}, month={Jan}, pages={77–102} }
 @article{liu_2016, title={On the commutative quotient of Fomin-Kirillov algebras}, volume={54}, ISSN={["1095-9971"]}, DOI={10.1016/j.ejc.2015.12.003}, abstractNote={The Fomin–Kirillov algebra En is a noncommutative algebra with a generator for each edge of the complete graph on n vertices. For any graph G on n vertices, let EG be the subalgebra of En generated by the edges in G. We show that the commutative quotient of EG is isomorphic to the Orlik–Terao algebra of G. As a consequence, the Hilbert series of this quotient is given by (−t)nχG(−t−1), where χG is the chromatic polynomial of G. We also give a reduction algorithm for the graded components of EG that do not vanish in the commutative quotient and show that their structure is described by the combinatorics of noncrossing forests.}, journal={EUROPEAN JOURNAL OF COMBINATORICS}, author={Liu, Ricky Ini}, year={2016}, month={May}, pages={65–75} }
 @article{blasiak_liu_meszaros_2016, title={Subalgebras of the Fomin-Kirillov algebra}, volume={44}, ISSN={["1572-9192"]}, DOI={10.1007/s10801-016-0688-4}, abstractNote={The Fomin–Kirillov algebra $${\mathcal {E}}_n$$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define $${{\mathcal {E}}_G}$$ to be the subalgebra of $${\mathcal {E}}_n$$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, $${\mathcal {E}}_G$$ is a free $${\mathcal {E}}_H$$ -module for any $$H\subseteq G$$ , and if $${\mathcal {E}}_G$$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for $${\mathcal {E}}_G$$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that $${\mathcal {E}}_G$$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of $${\mathcal {E}}_{\tilde{D}_n}$$ , $${\mathcal {E}}_{\tilde{E}_6}$$ , and $${\mathcal {E}}_{\tilde{E}_7}$$ , as well as the graphs G on six vertices for which $$\mathcal {E}_G$$ is finite-dimensional.}, number={3}, journal={JOURNAL OF ALGEBRAIC COMBINATORICS}, author={Blasiak, Jonah and Liu, Ricky Ini and Meszaros, Karola}, year={2016}, month={Nov}, pages={785–829} }
 @article{liu_2015, title={Positive expressions for skew divided difference operators}, volume={42}, ISSN={["1572-9192"]}, DOI={10.1007/s10801-015-0606-1}, abstractNote={For permutations $$v,w \in \mathfrak S_n$$ , Macdonald defines the skew divided difference operators $$\partial _{w/v}$$ as the unique linear operators satisfying $$\partial _w(PQ) = \sum _v v(\partial _{w/v}P) \cdot \partial _vQ$$ for all polynomials $$P$$ and $$Q$$ . We prove that $$\partial _{w/v}$$ has a positive expression in terms of divided difference operators $$\partial _{ij}$$ for $$i<j$$ . In fact, we prove that the analogous result holds in the Fomin–Kirillov algebra $${\mathcal {E}}_n$$ , which settles a conjecture of Kirillov.}, number={3}, journal={JOURNAL OF ALGEBRAIC COMBINATORICS}, author={Liu, Ricky Ini}, year={2015}, month={Nov}, pages={861–874} }
 @article{liu_2014, title={Coefficients of a relative of cyclotomic polynomials}, volume={165}, ISSN={0065-1036 1730-6264}, url={http://dx.doi.org/10.4064/aa165-4-1}, DOI={10.4064/aa165-4-1}, abstractNote={Let $N=p_1p_2... p_n$ be a product of $n$ distinct primes. Define $P_N(x)$ to be the polynomial $(1-x^N)\prod_{1\leq i<j\leq n}(1-x^{N/(p_ip_j)})/\prod_{i=1}^n (1-x^{N/p_i})$. (When $n=2$, $P_{pq}(x)$ is the $pq$-th cyclotomic polynomial, and when $n=3$, $P_{pqr}(x)$ is $(1-x)$ times the $pqr$-th cyclotomic polynomial.) Let the height of a polynomial be the maximum absolute value of one of its coefficients. It is well known that the height of $\Phi_{pq}(x)$ is 1, and Gallot and Moree showed that the same is true for $P_{pqr}(x)$ when $n=3$. We show that the coefficients of $P_N(x)$ depend mainly on the relative order of sums of residues of the form $p_j^{-1} \pmod {p_i}$. This allows us to explicitly describe the coefficients of $P_N(x)$ when $n=3$ and show that the height of $P_N(x)$ is at most 2 when $n=4$. We also show that for any $n$ there exist $P_N(x)$ with height 1 but that in general the maximum height of $P_N(x)$ is a function depending only on $n$ with growth rate $2^{n^2/2+O(n\log n)}$.}, number={4}, journal={Acta Arithmetica}, publisher={Institute of Mathematics, Polish Academy of Sciences}, author={Liu, Ricky Ini}, year={2014}, pages={301–325} }
 @article{liu_2014, title={Laurent polynomials, Eulerian numbers, and Bernstein's theorem}, volume={124}, ISSN={0097-3165}, url={http://dx.doi.org/10.1016/j.jcta.2014.02.003}, DOI={10.1016/j.jcta.2014.02.003}, abstractNote={Erman, Smith, and Várilly-Alvarado (2011) showed that the expected number of doubly monic Laurent polynomials f(z)=z−m+a−m+1z−m+1+⋯+an−1zn−1+zn whose first m+n−1 powers have vanishing constant term is the Eulerian number 〈m+n−1m−1〉, as well as a more refined result about sparse Laurent polynomials. We give an alternate proof of these results using Bernstein's theorem that clarifies the connection between these objects. In the process, we show that a refinement of Eulerian numbers gives a combinatorial interpretation for volumes of certain rational hyperplane sections of the hypercube.}, journal={Journal of Combinatorial Theory, Series A}, publisher={Elsevier BV}, author={Liu, Ricky Ini}, year={2014}, month={May}, pages={244–250} }
 @article{liu_2013, title={Nonconvexity of the set of hypergraph degree sequences}, volume={20}, number={1}, journal={Electronic Journal of Combinatorics}, author={Liu, Ricky Ini}, year={2013}, month={Jan} }
 @article{liu_2012, title={Matching polytopes and Specht modules}, volume={364}, ISSN={0002-9947 1088-6850}, url={http://dx.doi.org/10.1090/s0002-9947-2011-05516-9}, DOI={10.1090/s0002-9947-2011-05516-9}, abstractNote={We prove that the dimension of the Specht module of a forest $G$ is the same as the normalized volume of the matching polytope of $G$. We also associate to $G$ a symmetric function $s_G$ (analogous to the Schur symmetric function $s_\lambda$ for a partition $\lambda$) and investigate its combinatorial and representation-theoretic properties in relation to the Specht module and Schur module of $G$. We then use this to define notions of standard and semistandard tableaux for forests.}, number={2}, journal={Transactions of the American Mathematical Society}, publisher={American Mathematical Society (AMS)}, author={Liu, Ricky Ini}, year={2012}, month={Feb}, pages={1089–1107} }
 @article{lewis_liu_morales_panova_sam_zhang_2011, title={Matrices with restricted entries and $q$-analogues of permutations}, volume={2}, ISSN={2156-3527 2150-959X}, url={http://dx.doi.org/10.4310/joc.2011.v2.n3.a2}, DOI={10.4310/joc.2011.v2.n3.a2}, abstractNote={We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are $q$-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a $q$-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank, and we frame some of our results in the context of Lie theory. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.}, number={3}, journal={Journal of Combinatorics}, publisher={International Press of Boston}, author={Lewis, Joel Brewster and Liu, Ricky Ini and Morales, Alejandro H. and Panova, Greta and Sam, Steven V. and Zhang, Yan X.}, year={2011}, pages={355–395} }
 @article{liu_2010, title={An algorithmic Littlewood-Richardson rule}, volume={31}, ISSN={0925-9899 1572-9192}, url={http://dx.doi.org/10.1007/s10801-009-0184-1}, DOI={10.1007/s10801-009-0184-1}, abstractNote={We introduce a Littlewood-Richardson rule based on an algorithmic deformation of skew Young diagrams and present a bijection with the classical rule. The result is a direct combinatorial interpretation and proof of the geometric rule presented by Coskun (2000). We also present a corollary regarding the Specht modules of the intermediate diagrams.}, number={2}, journal={Journal of Algebraic Combinatorics}, publisher={Springer Science and Business Media LLC}, author={Liu, Ricky Ini}, year={2010}, month={Jan}, pages={253–266} }
 @phdthesis{liu_2010, title={Specht modules and Schubert varieties for general diagrams}, url={http://hdl.handle.net/1721.1/60196.}, journal={Thesis}, author={Liu, Ricky Ini}, year={2010}, month={May} }
 @article{liu_2007, title={Counting subrings of <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msup><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math> of index k}, volume={114}, ISSN={0097-3165}, url={http://dx.doi.org/10.1016/j.jcta.2006.05.002}, DOI={10.1016/j.jcta.2006.05.002}, abstractNote={We consider the problem of determining the number of subrings of the ring Z n of a fixed index k , denoted f n ( k ) . We present a decomposition theorem for these subrings and calculate explicit expressions for the Dirichlet series generating function F n ( s ) = ∑ k = 1 ∞ f n ( k ) k − s for n ⩽ 4 and for the generating function Φ p ( x , y ) = ∑ e = 0 ∞ ∑ n = 0 ∞ f n ( p e ) x e y n / n ! modulo p . We also calculate f n ( k ) when k is not divisible by the sixth power of any prime.}, number={2}, journal={Journal of Combinatorial Theory, Series A}, publisher={Elsevier BV}, author={Liu, Ricky Ini}, year={2007}, month={Feb}, pages={278–299} }
 @article{liu_2007, title={Counting subrings of Zn of index k}, volume={114}, number={2}, journal={Journal of Combinatorial Theory Series A}, author={Liu, Ricky Ini}, year={2007}, month={Feb}, pages={278–299} }