10 research outputs found

    Cores with distinct parts and bigraded Fibonacci numbers

    Full text link
    The notion of (a,b)(a,b)-cores is closely related to rational (a,b)(a,b) Dyck paths due to Anderson's bijection, and thus the number of (a,a+1)(a,a+1)-cores is given by the Catalan number CaC_a. Recent research shows that (a,a+1)(a,a+1) cores with distinct parts are enumerated by another important sequence- Fibonacci numbers FaF_a. In this paper, we consider the abacus description of (a,b)(a,b)-cores to introduce the natural grading and generalize this result to (a,as+1)(a,as+1)-cores. We also use the bijection with Dyck paths to count the number of (2k−1,2k+1)(2k-1,2k+1)-cores with distinct parts. We give a second grading to Fibonacci numbers, induced by bigraded Catalan sequence Ca,b(q,t)C_{a,b} (q,t)

    Quadratic ideals and Rogers-Ramanujan recursions

    Full text link
    We give an explicit recursive description of the Hilbert series and Gr\"obner bases for the family of quadratic ideals defining the jet schemes of a double point. We relate these recursions to the Rogers-Ramanujan identity and prove a conjecture of the second author, Oblomkov and Rasmussen.Comment: 17 page

    Combinatorial number theory through diagramming and gesture

    Get PDF
    Within combinatorial number theory, we study a variety of problems about whole numbers that include enumerative, diagrammatic, or computational elements. We present results motivated by two different areas within combinatorial number theory: the study of partitions and the study of digital representations of integers. We take the perspective that mathematics research is mathematics learning; existing research from mathematics education on mathematics learning and problem solving can be applied to mathematics research. We illustrate this by focusing on the concept of diagramming and gesture as mathematical practice. The mathematics presented is viewed through this lens throughout the document. Joint with H. E. Burson and A. Straub, motivated by recent results working toward classifying (s,t)(s, t)-core partitions into distinct parts, we present results on certain abaci diagrams. We give a recurrence (on ss) for generating polynomials for ss-core abaci diagrams with spacing dd and maximum position strictly less than ms−rms-r for positive integers ss, dd, mm, and rr. In the case r=1r =1, this implies a recurrence for (s,ms−1)(s, ms-1)-core partitions into dd-distinct parts, generalizing several recent results. We introduce the sets Q(b;{d1,d2,…,dk})Q(b;\{d_1, d_2, \ldots, d_k\}) to be integers that can be represented as quotients of integers that can be written in base bb using only digits from the set {d1,…,dk}\{d_1, \ldots, d_k\}. We explore in detail the sets Q(b;{d1,d2,…,dk})Q(b;\{d_1, d_2, \ldots, d_k\}) where d1=0d_1 = 0 and the remaining digits form proper subsets of the set {1,2,…,b−1}\{1, 2, \ldots, b-1\} for the cases b=3b =3, b=4b=4 and b=5b=5. We introduce modified multiplication transducers as a computational tool for studying these sets. We conclude with discussion of Q(b;{d1,…dk})Q(b; \{d_1, \ldots d_k\}) for general bb and digit sets including {−1,0,1}\{-1, 0, 1\}. Sections of this dissertation are written for a nontraditional audience (outside of the academic mathematics research community)
    corecore