39 research outputs found
A combinatorial proof of a plethystic Murnaghan--Nakayama rule
This article gives a combinatorial proof of a plethystic generalization of
the Murnaghan--Nakayama rule. The main result expresses the product of a Schur
function with the plethysm as an integral linear combination of
Schur functions. The proof uses a sign-reversing involution on sequences of
bead moves on James' abacus, inspired by the arguments in N. Loehr, Abacus
proofs of Schur function identities, SIAM J. Discrete Math. 24 (2010),
1356-1370.Comment: 9 pages, 2 figures, expanded and revised versio
Equivariant Quantum Cohomology of the Grassmannian via the Rim Hook Rule
A driving question in (quantum) cohomology of flag varieties is to find
non-recursive, positive combinatorial formulas for expressing the product of
two classes in a particularly nice basis, called the Schubert basis. Bertram,
Ciocan-Fontanine and Fulton provided a way to compute quantum products of
Schubert classes in the Grassmannian of k-planes in complex n-space by doing
classical multiplication and then applying a combinatorial rim hook rule which
yields the quantum parameter. In this paper, we provide a generalization of
this rim hook rule to the setting in which there is also an action of the
complex torus. Combining this result with Knutson and Tao's puzzle rule then
gives an effective algorithm for computing all equivariant quantum
Littlewood-Richardson coefficients. Interestingly, this rule requires a
specialization of torus weights modulo n, suggesting a direct connection to the
Peterson isomorphism relating quantum and affine Schubert calculus.Comment: 24 pages and 4 figures; typos corrected; final version to appear in
Algebraic Combinatoric
A Tile with Nested Chain Abacus
هذه الدراسة نجحت في ايجاد تمثيل جديد لمعداد James يدعى معداد السلاسل المتداخلة. المعداد الجديد وفر لنا تمثيل رياضي وحيد ل لتشفير اي بلاطه ( صوره ) باستخدام نظرية التجزئة بحيث ان كل شكل او صوره للبلاط سوف يقترن بتجزئة واحدة فقط بالإضافة الى هذا انشاء خوارزمية لحركة معداد السلاسل المتداخلة وهذه الحركة ممكن الاستفادة منها في نظرية التبليط.This study had succeeded in producing a new graphical representation of James abacus called nested chain abacus. Nested chain abacus provides a unique mathematical expression to encode each tile (image) using a partition theory where each form or shape of tile will be associated with exactly one partition.Furthermore, an algorithm of nested chain abacus movement will be constructed, which can be applied in tiling theory
A family of classes in nested chain abacus and related generating functions
Abacus model has been employed widely to represent partitions for any positive integer. However, no study has been carried out to develop connected beads of abacus in graphical representation for discrete objects. To resolve this connectedness problem this study is oriented in characterising n - connected objects knows as n connected ominoes, which then generate nested chain abacus. Furthermore, the theoretical conceptual properties for the nested chain abacus are being formulated. Along the construction, three different types of transformation are being created that are essential in building a family of classes. To enhance further, based on theses classes, generating functions are also being formulated by employing enumeration of combinatorial objects (ECO). In ECO method, each object is obtained from smaller object by making some local expansions. These local expansions are described in a simple way by a succession rule which can be translated into a function equation for the generating function. In summary, this stud has succeeded in producing novel graphical representation of nested chain abacus, which can be applied in tiling finite grid