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  • 1.
    Alexandersson, Per
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Polytopes and Large Counterexamples2019Inngår i: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 28, nr 1, s. 115-120Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this short note, we give large counterexamples to natural questions about certain order polytopes, in particular, Gelfand–Tsetlin polytopes. Several of the counterexamples are too large to be discovered via a brute-force computer search. We also show that the multiset of hooks in a Young diagram is not enough information to determine the Ehrhart polynomial for an associated order polytope. This is somewhat counter-intuitive to the fact that the multiset of hooks always determine the leading coefficient of the Ehrhart polynomial.

  • 2.
    Alexandersson, Per
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Jordan, Linus
    Ecole Polytech Fed Lausanne, Dept Math, CH-1015 Lausanne, Switzerland..
    Enumeration of Border-Strip Decompositions and Weil-Petersson Volumes2019Inngår i: Journal of Integer Sequences, ISSN 1530-7638, E-ISSN 1530-7638, Vol. 22, nr 4, artikkel-id 19.4.5Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We describe an injection from border-strip decompositions of certain diagrams to permutations. This allows us to provide enumeration results as well as q-analogues of enumeration formulas. Finally, we use this injection to prove a connection between the number of border-strip decompositions of the n x 2n rectangle and the Weil-Petersson volume of the moduli space of an n-punctured Riemann sphere.

  • 3.
    Alexandersson, Per
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Panova, Greta
    LLT polynomials, chromatic quasisymmetric functions and graphs with cycles2018Inngår i: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 341, nr 12, s. 3453-3482Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as unicellular LLT polynomials, revealing some parallel structure and phenomena regarding their e-positivity. The Dyck path model is also extended to circular arc digraphs to obtain larger families of polynomials, giving a new extension of LLT polynomials. Carrying over a lot of the noncircular combinatorics, we prove several statements regarding the e-coefficients of chromatic quasisymmetric functions and LLT polynomials, including a natural combinatorial interpretation for the e-coefficients for the line graph and the cycle graph for both families. We believe that certain e-positivity conjectures hold in all these families above. Furthermore, beyond the chromatic analogy, we study vertical-strip LLT polynomials, which are modified Hall-Littlewood polynomials. 

  • 4.
    Alexandersson, Per
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Sawhney, Mehtaab
    A Major-Index Preserving Map on Fillings2017Inngår i: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 24, nr 4, artikkel-id P4.3Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We generalize a map by S. Mason regarding two combinatorial models for key polynomials, in a way that accounts for the major index. Furthermore we define a similar variant of this map, that regards alternative models for the modified Macdonald polynomials at t = 0, and thus partially answers a question by J. Haglund. These maps together imply a certain uniqueness property regarding inversion- and coinversion-free fillings. These uniqueness properties allow us to generalize the notion of charge to a non-symmetric setting, thus answering a question by A. Lascoux and the analogous question in the symmetric setting proves a conjecture by K. Nelson.

  • 5.
    Amini, Nima
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Alexandersson, Per
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    The cone of cyclic sieving phenomena2019Inngår i: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 342, nr 6, s. 1581-1601Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We study cyclic sieving phenomena (CSP) on combinatorial objects from an abstract point of view by considering a rational polyhedral cone determined by the linear equations that define such phenomena. Each lattice point in the cone corresponds to a non-negative integer matrix which jointly records the statistic and cyclic order distribution associated with the set of objects realizing the CSP. In particular we consider a universal subcone onto which every CSP matrix linearly projects such that the projection realizes a CSP with the same cyclic orbit structure, but via a universal statistic that has even distribution on the orbits.

    Reiner et.al. showed that every cyclic action gives rise to a unique polynomial (mod q^n-1) complementing the action to a CSP. We give a necessary and sufficient criterion for the converse to hold. This characterization allows one to determine if a combinatorial set with a statistic gives rise (in principle) to a CSP without having a combinatorial realization of the cyclic action. We apply the criterion to conjecture a new CSP involving stretched Schur polynomials and prove our conjecture for certain rectangular tableaux. Finally we study some geometric properties of the CSP cone. We explicitly determine its half-space description and in the prime order case we determine its extreme rays.

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