5 research outputs found
Shifted distinct-part partition identities in arithmetic progressions
The partition function , which counts the number of partitions of a
positive integer , is widely studied. Here, we study partition functions
that count partitions of into distinct parts satisfying certain
congruence conditions. A shifted partition identity is an identity of the form
for all in some arithmetic progression. Several
identities of this type have been discovered, including two infinite families
found by Alladi. In this paper, we use the theory of modular functions to
determine the necessary and sufficient conditions for such an identity to
exist. In addition, for two specific cases, we extend Alladi's theorem to other
arithmetic progressions
Coincidences among skew Grothendieck polynomials
The question of when two skew Young diagrams produce the same skew Schur function has been well-studied. We investigate the same question in the case of stable Grothendieck polynomials, which are the K-theoretic analogues of the Schur functions. We prove a necessary condition for two skew shapes to give rise to the same dual stable Grothendieck polynomial. We also provide a necessary and sufficient condition in the case where the two skew shapes are ribbons