@article{hersh_lenart_2017, title={From the weak Bruhat order to crystal posets}, volume={286}, ISSN={["1432-1823"]}, DOI={10.1007/s00209-016-1808-5}, abstractNote={We investigate the ways in which fundamental properties of the weak Bruhat order on a Weyl group can be lifted (or not) to a corresponding highest weight crystal graph, viewed as a partially ordered set; the latter projects to the weak order via the key map. First, a crystal theoretic analogue of the statement that any two reduced expressions for the same Coxeter group element are related by Coxeter moves is proven for all lower intervals $$[\hat{0},v]$$ in a simply or doubly laced crystal. On the other hand, it is shown that no finite set of moves exists, even in type A, for arbitrary crystal graph intervals. In fact, it is shown that there are relations of arbitrarily high degree amongst crystal operators that are not implied by lower degree relations. Second, for crystals associated to Kac–Moody algebras it is shown for lower intervals that the Möbius function is always 0 or ±1, and in finite type this is also proven for upper intervals, with a precise formula given in each case. Moreover, the order complex for each of these intervals is proven to be homotopy equivalent to a ball or to a sphere of some dimension, despite often not being shellable. For general intervals, examples are constructed with arbitrarily large Möbius function, again even in type A. Any interval having Möbius function other than 0 or ±1 is shown to contain within it a relation amongst crystal operators that is not implied by the relations giving rise to the local structure of the crystal, making precise a tight relationship between the Möbius function and these somewhat unexpected relations appearing in crystals. New properties of the key map are also derived. The key is shown to be determined entirely by the edge-colored poset-theoretic structure of the crystal, and a recursive algorithm is given for calculating it. In finite types, the fiber of the longest element of any parabolic subgroup of the Weyl group is also proven to have a unique minimal and a unique maximal element; this property fails for more general elements of the Weyl group.}, number={3-4}, journal={MATHEMATISCHE ZEITSCHRIFT}, author={Hersh, Patricia and Lenart, Cristian}, year={2017}, month={Aug}, pages={1435–1464} }
 @article{hersh_reiner_2017, title={Representation Stability for Cohomology of Configuration Spaces in R-d}, volume={2017}, ISSN={["1687-0247"]}, DOI={10.1093/imrn/rnw060}, abstractNote={This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group $S_n$ on the cohomology of the configuration space of $n$ ordered points in $\mathbf{R}^d$. This cohomology is known to vanish outside of dimensions divisible by $d-1$; it is shown here that the $S_n$-representation on the $i(d-1)^{st}$ cohomology stabilizes sharply at $n=3i$ (resp. $n=3i+1$) when $d$ is odd (resp. even). 
The result comes from analyzing $S_n$-representations known to control the cohomology: the Whitney homology of set partition lattices for $d$ even, and the higher Lie representations for $d$ odd. A similar analysis shows that the homology of any rank-selected subposet in the partition lattice stabilizes by $n\geq 4i$, where $i$ is the maximum rank selected. 
Further properties of the Whitney homology and more refined stability statements for $S_n$-isotypic components are also proven, including conjectures of J. Wiltshire-Gordon.}, number={5}, journal={INTERNATIONAL MATHEMATICS RESEARCH NOTICES}, author={Hersh, Patricia and Reiner, Victor}, year={2017}, month={Mar}, pages={1433–1486} }
 @article{hersh_meszaros_2017, title={SB-labelings and posets with each interval homotopy equivalent to a sphere or a ball}, volume={152}, ISSN={["1096-0899"]}, DOI={10.1016/j.jcta.2017.06.001}, abstractNote={We introduce a new class of edge labelings for locally finite lattices which we call SB-labelings. We prove for finite lattices which admit an SB-labeling that each open interval has the homotopy type of a ball or of a sphere of some dimension. Natural examples include the weak order, the Tamari lattice, and the finite distributive lattices.}, journal={JOURNAL OF COMBINATORIAL THEORY SERIES A}, author={Hersh, Patricia and Meszaros, Karola}, year={2017}, month={Nov}, pages={104–120} }
 @article{davidson_hersh_2014, title={A lexicographic shellability characterization of geometric lattices}, volume={123}, ISSN={["1096-0899"]}, DOI={10.1016/j.jcta.2013.11.001}, abstractNote={Geometric lattices are characterized in this paper as those finite, atomic lattices such that every atom ordering induces a lexicographic shelling given by an edge labeling known as a minimal labeling. Equivalently, geometric lattices are shown to be exactly those finite lattices such that every ordering on the join-irreducibles induces a lexicographic shelling. This new characterization fits into a similar paradigm as McNamaraʼs characterization of supersolvable lattices as those lattices admitting a different type of lexicographic shelling, namely one in which each maximal chain is labeled with a permutation of {1,…,n}.}, number={1}, journal={JOURNAL OF COMBINATORIAL THEORY SERIES A}, author={Davidson, Ruth and Hersh, Patricia}, year={2014}, month={Apr}, pages={8–13} }
 @article{hersh_2014, title={Regular cell complexes in total positivity}, volume={197}, ISSN={["1432-1297"]}, DOI={10.1007/s00222-013-0480-1}, abstractNote={Fomin and Shapiro conjectured that the link of the identity in the Bruhat stratification of the totally nonnegative real part of the unipotent radical of a Borel subgroup in a semisimple, simply connected algebraic group defined and split over ${\mathbb{R}}$ is a regular CW complex homeomorphic to a ball. The main result of this paper is a proof of this conjecture. This completes the solution of the question of Bernstein of identifying regular CW complexes arising naturally from representation theory having the (lower) intervals of Bruhat order as their closure posets. A key ingredient is a new criterion for determining whether a finite CW complex is regular with respect to a choice of characteristic maps; it most naturally applies to images of maps from regular CW complexes and is based on an interplay of combinatorics of the closure poset with codimension one topology.}, number={1}, journal={INVENTIONES MATHEMATICAE}, author={Hersh, Patricia}, year={2014}, month={Jul}, pages={57–114} }
 @article{hersh_shareshian_stanton_2014, title={The q = -1 phenomenon via homology concentration}, volume={5}, ISSN={2156-3527 2150-959X}, url={http://dx.doi.org/10.4310/joc.2014.v5.n2.a2}, DOI={10.4310/joc.2014.v5.n2.a2}, abstractNote={We introduce a homological approach to exhibiting instances of Stembridge's q=-1 phenomenon. This approach is shown to explain two important instances of the phenomenon, namely that of partitions whose Ferrers diagrams fit in a rectangle of fixed size and that of plane partitions fitting in a box of fixed size. A more general framework of invariant and coinvariant complexes with coefficients taken mod 2 is developed, and as a part of this story an analogous homological result for necklaces is conjectured.}, number={2}, journal={Journal of Combinatorics}, publisher={International Press of Boston}, author={Hersh, P. and Shareshian, J. and Stanton, D.}, year={2014}, pages={167–194} }
 @article{hersh_schilling_2013, title={Symmetric Chain Decomposition for Cyclic Quotients of Boolean Algebras and Relation to Cyclic Crystals}, volume={2013}, ISSN={["1687-0247"]}, DOI={10.1093/imrn/rnr254}, abstractNote={The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of the $\mathfrak{sl}_2$ lowering operator in the theory of crystal bases.}, number={2}, journal={INTERNATIONAL MATHEMATICS RESEARCH NOTICES}, author={Hersh, Patricia and Schilling, Anne}, year={2013}, pages={463–473} }
 @article{engström_hersh_sturmfels_2013, title={Toric cubes}, volume={62}, ISSN={0009-725X 1973-4409}, url={http://dx.doi.org/10.1007/s12215-013-0115-9}, DOI={10.1007/s12215-013-0115-9}, number={1}, journal={Rendiconti del Circolo Matematico di Palermo}, publisher={Springer Science and Business Media LLC}, author={Engström, Alexander and Hersh, Patricia and Sturmfels, Bernd}, year={2013}, month={Feb}, pages={67–78} }
 @article{armstrong_hersh_2011, title={Sorting orders, subword complexes, Bruhat order and total positivity}, volume={46}, ISSN={["1090-2074"]}, DOI={10.1016/j.aam.2010.09.006}, abstractNote={In this note we construct a poset map from a Boolean algebra to the Bruhat order which unveils an interesting connection between subword complexes, sorting orders, and certain totally nonnegative spaces. This relationship gives a simple new proof that the proper part of Bruhat order is homotopy equivalent to the proper part of a Boolean algebra — that is, to a sphere. We also obtain a geometric interpretation for sorting orders. We conclude with two new results: that the intersection of all sorting orders is the weak order, and the union of sorting orders is the Bruhat order.}, number={1-4}, journal={ADVANCES IN APPLIED MATHEMATICS}, author={Armstrong, Drew and Hersh, Patricia}, year={2011}, month={Jan}, pages={46–53} }
 @article{hersh_lenart_2010, title={Combinatorial constructions of weight bases: The Gelfand-Tsetlin basis}, volume={17}, number={1}, journal={Electronic Journal of Combinatorics}, author={Hersh, P. and Lenart, C.}, year={2010} }
 @inbook{fienberg_hersh_rinaldo_zhou_2010, title={Maximum likelihood estimation in latent class models for contingency table data}, ISBN={9780511642401}, url={http://dx.doi.org/10.1017/cbo9780511642401.003}, DOI={10.1017/cbo9780511642401.003}, abstractNote={Statistical models with latent structure have a history going back to the 1950s and have seen widespread use in the social sciences and, more recently, in computational biology and in machine learning. Here we study the basic latent class model proposed originally by the sociologist Paul F. Lazarfeld for categorical variables, and we explain its geometric structure. We draw parallels between the statistical and geometric properties of latent class models and we illustrate geometrically the causes of many problems associated with maximum likelihood estimation and related statistical inference. In particular, we focus on issues of non-identifiability and determination of the model dimension, of maximisation of the likelihood function and on the effect of symmetric data. We illustrate these phenomena with a variety of synthetic and real-life tables, of different dimension and complexity. Much of the motivation for this work stems from the ‘100 Swiss Francs’ problem, which we introduce and describe in detail.}, booktitle={Algebraic and Geometric Methods in Statistics}, publisher={Cambridge University Press}, author={Fienberg, S. E. and Hersh, P. and Rinaldo, A. and Zhou, Y.}, editor={Gibilisco, Paolo and Riccomagno, Eva and Rogantin, Maria Piera and Wynn, Henry P.Editors}, year={2010}, month={Jul}, pages={27–62} }
 @misc{hersh_kleinberg_2009, title={A multiplicative deformation of the Möbius function for the poset of partitions of a multiset}, ISBN={9780821843451 9780821881583}, ISSN={1098-3627 0271-4132}, url={http://dx.doi.org/10.1090/conm/479/09346}, DOI={10.1090/conm/479/09346}, abstractNote={The Möbius function of a partially ordered set is a very convenient formalism for counting by inclusion-exclusion. An example of central importance is the partition lattice, namely the partial order by refinement on partitions of a set {1, . . . , n}. It seems quite natural to generalize this to partitions of a multiset, i.e. to allow repetition of the letters. However, the Möbius function is not nearly so well-behaved. We introduce a multiplicative deformation, denoted μ, for the Möbius function of the poset of partitions of a multiset and show that it possesses much more elegant formulas than the usual Möbius function in this case.}, journal={Contemporary Mathematics}, publisher={American Mathematical Society}, author={Hersh, Patricia and Kleinberg, Robert}, year={2009}, pages={113–118} }
 @article{hersh_hsiao_2009, title={Random walks on quasisymmetric functions}, volume={222}, ISSN={0001-8708}, url={http://dx.doi.org/10.1016/j.aim.2009.05.014}, DOI={10.1016/j.aim.2009.05.014}, abstractNote={Conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations. Spectral results are also obtained. Several important random walks are now realized this way: Stanley's QS-distribution results from endomorphisms given by evaluation maps, a-shuffles result from the ath convolution power of the universal character, and the Tchebyshev operator of the second kind introduced recently by Ehrenborg and Readdy yields traditional riffle shuffles. A conjecture of Ehrenborg regarding the spectra for a family of random walks on ab-words is proven. A theorem of Stembridge from the theory of enriched P-partitions is also recovered as a special case.}, number={3}, journal={Advances in Mathematics}, publisher={Elsevier BV}, author={Hersh, Patricia and Hsiao, Samuel K.}, year={2009}, month={Oct}, pages={782–808} }
 @article{hersh_2009, title={Shelling Coxeter-like complexes and sorting on trees}, volume={221}, ISSN={["0001-8708"]}, DOI={10.1016/j.aim.2009.01.007}, abstractNote={In their work on ‘Coxeter-like complexes’, Babson and Reiner introduced a simplicial complex ΔT associated to each tree T on n nodes, generalizing chessboard complexes and type A Coxeter complexes. They conjectured that ΔT is (n−b−1)-connected when the tree has b leaves. We provide a shelling for the (n−b)-skeleton of ΔT, thereby proving this conjecture. In the process, we introduce notions of weak order and inversion functions on the labellings of a tree T which imply shellability of ΔT, and we construct such inversion functions for a large enough class of trees to deduce the aforementioned conjecture and also recover the shellability of chessboard complexes Mm,n with n⩾2m−1. We also prove that the existence or nonexistence of an inversion function for a fixed tree governs which networks with a tree structure admit greedy sorting algorithms by inversion elimination and provide an inversion function for trees where each vertex has capacity at least its degree minus one.}, number={3}, journal={ADVANCES IN MATHEMATICS}, author={Hersh, Patricia}, year={2009}, month={Jun}, pages={812–829} }
 @article{hersh_shareshian_stanton_2009, title={The q=-1 phenomenon for bounded (plane) partitions via homology concentration}, journal={Discrete Mathematics and Theoretical Computer Science}, author={Hersh, Patricia and Shareshian, John and Stanton, Dennis}, year={2009}, pages={471–484} }
 @article{hersh_shareshian_2007, title={Chains of Modular Elements and Lattice Connectivity}, volume={23}, ISSN={0167-8094 1572-9273}, url={http://dx.doi.org/10.1007/s11083-006-9053-x}, DOI={10.1007/s11083-006-9053-x}, number={4}, journal={Order}, publisher={Springer Science and Business Media LLC}, author={Hersh, Patricia and Shareshian, John}, year={2007}, month={Jan}, pages={339–342} }
 @article{hersh_swartz_2007, title={Coloring complexes and arrangements}, volume={27}, ISSN={0925-9899 1572-9192}, url={http://dx.doi.org/10.1007/s10801-007-0086-z}, DOI={10.1007/s10801-007-0086-z}, abstractNote={Steingrimsson's coloring complex and Jonsson's unipolar complex are interpreted in terms of hyperplane arrangements. This viewpoint leads to short proofs that all coloring complexes and a large class of unipolar complexes have convex ear decompositions. These convex ear decompositions impose strong new restrictions on the chromatic polynomials of all finite graphs. Similar results are obtained for characteristic polynomials of submatroids of type ℬ n arrangements.}, number={2}, journal={Journal of Algebraic Combinatorics}, publisher={Springer Science and Business Media LLC}, author={Hersh, Patricia and Swartz, Ed}, year={2007}, month={Jul}, pages={205–214} }
 @article{berglund_blasiak_hersh_2007, title={Combinatorics of multigraded Poincaré series for monomial rings}, volume={308}, ISSN={0021-8693}, url={http://dx.doi.org/10.1016/j.jalgebra.2006.08.020}, DOI={10.1016/j.jalgebra.2006.08.020}, abstractNote={Backelin proved that the multigraded Poincaré series for resolving a residue field over a polynomial ring modulo a monomial ideal is a rational function. The numerator is simple, but until the recent work of Berglund there was no combinatorial formula for the denominator. Berglund's formula gives the denominator in terms of ranks of reduced homology groups of lower intervals in a certain lattice. We now express this lattice as the intersection lattice LA(I) of a subspace arrangement A(I), use Crapo's Closure Lemma to drastically simplify the denominator in some cases (such as monomial ideals generated in degree two), and relate Golodness to the Cohen–Macaulay property for associated posets. In addition, we introduce a new class of finite lattices called complete lattices, prove that all geometric lattices are complete and provide a simple criterion for Golodness of monomial ideals whose lcm-lattices are complete.}, number={1}, journal={Journal of Algebra}, publisher={Elsevier BV}, author={Berglund, Alexander and Blasiak, Jonah and Hersh, Patricia}, year={2007}, month={Feb}, pages={73–90} }
 @article{babson_hersh_2005, volume={357}, ISSN={0002-9947}, url={http://dx.doi.org/10.1090/s0002-9947-04-03495-6}, DOI={10.1090/s0002-9947-04-03495-6}, abstractNote={This paper shows how to construct a discrete Morse function with a relatively small number of critical cells for the order complex of any finite poset with O and I from any lexicographic order on its maximal chains. Specifically, if we attach facets according to the lexicographic order on maximal chains, then each facet contributes at most one new face which is critical, and at most one Betti number changes; facets which do not change the homotopy type also do not contribute any critical faces. Dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets. As one application, the Mobius function may be computed as the alternating sum of Morse numbers. The above construction enables us to prove that the poset Π n /S λ of partitions of a set {1 λ 1,...,k λ k} with repetition is homotopy equivalent to a wedge of spheres of top dimension when A is a hook-shaped partition; it is likely that the proof may be extended to a larger class of A and perhaps to all A, despite a result of Ziegler (1986) which shows that Π n /S λ is not always Cohen-Macaulay.}, number={02}, journal={Transactions of the American Mathematical Society}, publisher={American Mathematical Society (AMS)}, author={Babson, Eric and Hersh, Patricia}, year={2005}, month={Feb}, pages={509–535} }
 @inbook{hersh_welker_2005, place={Providence, RI}, series={Contemporary Mathematics}, title={Gröbner basis degree bounds on Tor•^K[Λ](k, k)• and discrete Morse theory for posets}, ISBN={9780821834596 9780821879641}, DOI={10.1090/conm/374/06902}, booktitle={Integer Points in Polyhedra-geometry, number theory, algebra, optimization}, publisher={American Mathematical Society}, author={Hersh, Patricia and Welker, Volkmar}, editor={Barvinok, Alexander and Beck, Matthias and Haase, Christian and Reznick, Bruce and Welker, VolkmarEditors}, year={2005}, pages={101–138}, collection={Contemporary Mathematics} }
 @article{hersh_2005, title={On optimizing discrete Morse functions}, volume={35}, ISSN={0196-8858}, url={http://dx.doi.org/10.1016/j.aam.2005.04.001}, DOI={10.1016/j.aam.2005.04.001}, abstractNote={In 1998, Forman introduced discrete Morse theory as a tool for studying CW complexes by producing smaller, simpler-to-understand complexes of critical cells with the same homotopy types as the original complexes. This paper addresses two questions: (1) under what conditions may several gradient paths in a discrete Morse function simultaneously be reversed to cancel several pairs of critical cells, to further collapse the complex, and (2) which gradient paths are individually reversible in lexicographic discrete Morse functions on poset order complexes. The latter follows from a correspondence between gradient paths and lexicographically first reduced expressions for permutations. As an application, a new partial order on the symmetric group recently introduced by Remmel is proven to be Cohen–Macaulay.}, number={3}, journal={Advances in Applied Mathematics}, publisher={Elsevier BV}, author={Hersh, Patricia}, year={2005}, month={Sep}, pages={294–322} }
 @article{hanlon_hersh_2004, title={A Hodge decomposition for the complex of injective words}, volume={214}, ISSN={0030-8730}, url={http://dx.doi.org/10.2140/pjm.2004.214.109}, DOI={10.2140/pjm.2004.214.109}, abstractNote={llReiner and Webb (preprint, 2002) compute the S n -module structure for the complex of injective words. This paper refines their formula by providing a Hodge type decomposition. Along the way, this paper proves that the simplicial boundary map interacts in a nice fashion with the Eulerian idempotents. The Laplacian acting on the top chain group in the complex of injective words is also shown to equal the signed random to random shuffle operator. Uyemura-Reyes, 2002, conjectured that the (unsigned) random to random shuffle operator has integral spectrum. We prove that this conjecture would imply that the Laplacian on (each chain group in) the complex of injective words has integral spectrum.}, number={1}, journal={Pacific Journal of Mathematics}, publisher={Mathematical Sciences Publishers}, author={Hanlon, Phil and Hersh, Patricia}, year={2004}, month={Mar}, pages={109–125} }
 @article{hersh_2004, title={Connectivity of h-complexes}, volume={105}, ISSN={0097-3165}, url={http://dx.doi.org/10.1016/j.jcta.2003.10.006}, DOI={10.1016/j.jcta.2003.10.006}, abstractNote={This paper verifies a conjecture of Edelman and Reiner regarding the homology of the h -complex of a Boolean algebra. A discrete Morse function with no low-dimensional critical cells is constructed, implying a lower bound on connectivity. This together with an Alexander duality result of Edelman and Reiner implies homology vanishing also in high dimensions. Finally, possible generalizations to certain classes of supersolvable lattices are suggested.}, number={1}, journal={Journal of Combinatorial Theory, Series A}, publisher={Elsevier BV}, author={Hersh, Patricia}, year={2004}, month={Jan}, pages={111–126} }
 @article{hersh_2003, title={A partitioning and related properties for the quotient complex Δ(Blm)/Sl≀Sm}, volume={178}, ISSN={0022-4049}, url={http://dx.doi.org/10.1016/s0022-4049(02)00192-5}, DOI={10.1016/s0022-4049(02)00192-5}, abstractNote={We study the quotient complex Δ(Blm)/Sl≀Sm as a means of deducing facts about the ring k[x1,…,xlm]Sl≀Sm. It is shown in Hersh (preprint, 2000) that Δ(Blm)/Sl≀Sm is shellable when l=2, implying Cohen–Macaulayness of k[x1,…,x2m]S2≀Sm for any field k. We now confirm for all pairs (l,m) with l>2 and m>1 that Δ(Blm)/Sl≀Sm is not Cohen–Macaulay over Z/2Z, but it is Cohen–Macaulay over fields of characteristic p>m (independent of l). This yields corresponding characteristic-dependent results for k[x1,…,xlm]Sl≀Sm. We also prove that Δ(Blm)/Sl≀Sm and the links of many of its faces are collapsible, and we give a partitioning for Δ(Blm)/Sl≀Sm.}, number={3}, journal={Journal of Pure and Applied Algebra}, publisher={Elsevier BV}, author={Hersh, Patricia}, year={2003}, month={Mar}, pages={255–272} }
 @article{hersh_2003, title={Chain decomposition and the flag f-vector}, volume={103}, ISSN={0097-3165}, url={http://dx.doi.org/10.1016/s0097-3165(03)00066-9}, DOI={10.1016/s0097-3165(03)00066-9}, abstractNote={Ehrenborg introduced a quasi-symmetric function encoding, denoted FP, for the flag f-vector of any finite, graded poset P with 0̂ and 1̂. Stanley observed that FP is a symmetric function whenever P is locally rank-symmetric and asked for conditions under which FP is Schur-positive. We provide formulas for FP for three classes of locally rank-symmetric posets: graded monoid posets, generalized posets of shuffles and noncrossing partition lattices for classical reflection groups. Our flag f-vector expressions for generalized shuffle posets and noncrossing partition lattices exhibit Schur-positivity, while graded monoid posets do not always have Schur-positive flag f-vector. Each of our flag f-vector expressions results from a poset chain decomposition. For the noncrossing partition lattices and shuffle posets, these also yield symmetric chain decompositions (by restriction to 1-chains), shellability and supersolvability results and combinatorial formulae including characteristic polynomial and zeta polynomial. Our (more complicated) flag f-vector expression for graded monoid posets involves Gröbner bases and a weighted notion of Möbius function for the poset of partitions of a multiset and related multiset intersection posets.}, number={1}, journal={Journal of Combinatorial Theory, Series A}, publisher={Elsevier BV}, author={Hersh, Patricia}, year={2003}, month={Jul}, pages={27–52} }
 @article{hersh_2003, title={Lexicographic Shellability for Balanced Complexes}, volume={17}, ISSN={0925-9899}, url={http://dx.doi.org/10.1023/a:1025044720847}, DOI={10.1023/a:1025044720847}, abstractNote={We introduce a notion of lexicographic shellability for pure, balanced boolean cell complexes, modelled after the CL-shellability criterion of Björner and Wachs (Adv. in Math. 43 (1982), 87–100) for posets and its generalization by Kozlov (Ann. of Comp. 1(1) (1997), 67–90) called CC-shellability. We give a lexicographic shelling for the quotient of the order complex of a Boolean algebra of rank 2n by the action of the wreath product S 2 ≀ S n of symmetric groups, and we provide a partitioning for the quotient complex Δ(Π n )/S n. Stanley asked for a description of the symmetric group representation β S on the homology of the rank-selected partition lattice Πn S in Stanley (J. Combin. Theory Ser. A 32(2) (1982), 132–161), and in particular he asked when the multiplicity b S(n) of the trivial representation in β S is 0. One consequence of the partitioning for Δ(Π n )/S n is a (fairly complicated) combinatorial interpretation for b S(n); another is a simple proof of Hanlon's result (European J. Combin. 4(2) (1983), 137–141) that b 1,⋯,i(n) = 0. Using a result of Garsia and Stanton from (Adv. in Math. 51(2) (1984), 107–201), we deduce from our shelling for Δ(B 2n )/S 2 ≀ S n that the ring of invariants k[x 1,⋯,x 2 n ] S2 ≀ Sn is Cohen-Macaulay over any field k.}, number={3}, journal={Journal of Algebraic Combinatorics}, publisher={Springer Nature}, author={Hersh, Patricia}, year={2003}, month={May}, pages={225–254} }
 @article{hanlon_hersh_2003, title={Multiplicity of the trivial representation in rank-selected homology of the partition lattice}, volume={266}, ISSN={0021-8693}, url={http://dx.doi.org/10.1016/s0021-8693(03)00372-7}, DOI={10.1016/s0021-8693(03)00372-7}, abstractNote={We study the multiplicity bS(n) of the trivial representation in the symmetric group representations βS on the (top) homology of the rank-selected partition lattice ΠnS. We break the possible rank sets S into three cases: (1) 1∉S, (2) S=1,…,i for i⩾1, and (3) S=1,…,i,j1,…,jl for i,l⩾1, j1>i+1. It was previously shown by Hanlon that bS(n)=0 for S=1,…,i. We use a partitioning for Δ(Πn)/Sn due to Hersh to confirm a conjecture of Sundaram [S. Sundaram, The homology representations of the symmetric group on Cohen–Macaulay subposets of the partition lattice, Adv. Math. 104 (1994) 225–296] that bS(n)>0 for 1∉S. On the other hand, we use the spectral sequence of a filtered complex to show bS(n)=0 for S=1,…,i,j1,…,jl unless a certain type of chain of support S exists. The partitioning for Δ(Πn)/Sn allows us then to show that a large class of rank sets S=1,…,i,j1,…,jl for which such a chain exists do satisfy bS(n)>0. We also generalize the partitioning for Δ(Πn)/Sn to Δ(Πn)/Sλ; when λ=(n−1,1), this partitioning leads to a proof of a conjecture of Sundaram about (S1×Sn−1)-representations on the homology of the partition lattice.}, number={2}, journal={Journal of Algebra}, publisher={Elsevier BV}, author={Hanlon, Phil and Hersh, Patricia}, year={2003}, month={Aug}, pages={521–538} }
 @article{hersh_novik_2002, title={A Short Simplicial h -Vector and the Upper Bound Theorem}, volume={28}, ISSN={0179-5376 1432-0444}, url={http://dx.doi.org/10.1007/s00454-002-0746-7}, DOI={10.1007/s00454-002-0746-7}, abstractNote={The Upper Bound Conjecture is verified for a class of odd-dimensional simplicial complexes that in particular includes all Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes—a short simplicial h -vector.}, number={3}, journal={Discrete & Computational Geometry}, publisher={Springer Science and Business Media LLC}, author={Hersh, Patricia and Novik, Isabella}, year={2002}, month={Aug}, pages={283–289} }
 @article{hersh_2002, title={Two Generalizations of Posets of Shuffles}, volume={97}, ISSN={0097-3165}, url={http://dx.doi.org/10.1006/jcta.2001.3187}, DOI={10.1006/jcta.2001.3187}, abstractNote={We study posets defined by Stanley as a multiset generalization of Greene's posets of shuffles. Ehrenborg defined a quasi-symmetric function encoding for the flag f-vector, denoted FP, and we determine FP for shuffle posets of multisets, expressing it as a Schur-positive symmetric function. This leads to several combinatorial formulas as well as proofs that shuffle posets of multisets are supersolvable and have symmetric chain decompositions. We also generalize posets of shuffles to posets for shuffling k words, answering a question of Stanley. Finally, we extend our results about shuffle posets of multisets to k-shuffle posets.}, number={1}, journal={Journal of Combinatorial Theory, Series A}, publisher={Elsevier BV}, author={Hersh, Patricia}, year={2002}, month={Jan}, pages={1–26} }
 @phdthesis{hersh_1999, title={Decomposition and Enumeration in Partially Ordered Sets}, school={Massachusetts Institute of Technology}, author={Hersh, Patricia}, year={1999}, month={May} }
 @article{hersh_1999, title={Deformation of chains via a local symmetric group action}, volume={R27}, journal={The Electronic Journal of Combinatorics}, author={Hersh, Patricia}, year={1999} }
 @article{hersh_1999, title={On exact n-step domination}, volume={205}, ISSN={0012-365X}, url={http://dx.doi.org/10.1016/s0012-365x(99)00024-2}, DOI={10.1016/s0012-365x(99)00024-2}, abstractNote={We generalize to n steps the notion of exact 2-step domination introduced by Chartrand et al. (Math. Bohem. 120 (1995) 125–134) and suggest a related minimization problem for which we find a lower bound. A graph G is an exact n -step domination graph if there is some set of vertices in G such that each vertex in the graph is distance n from exactly one vertex in the set. We prove that such subsets have order at least ⌊ log 2 n⌋+2 and limit how much better a bound is possible. We also prove a related conjecture of Alavi et al. (Graph Theory, Combinatorics, and Applications, vol. 1, Wiley, New York, 1991, pp. 1–8) that if each vertex in a connected graph G has exactly one vertex distance n from it then the diameter is n unless G is a path consisting of 2 n vertices.}, number={1-3}, journal={Discrete Mathematics}, publisher={Elsevier BV}, author={Hersh, Patricia}, year={1999}, month={Jul}, pages={235–239} }