43 research outputs found
The BG-rank of a partition and its applications
Let \pi be a partition. In [2] we defined BG-rank(\pi) as an alternating sum
of parities of parts. This statistic was employed to generalize and refine the
famous Ramanujan modulo 5 partition congruence. Let p_j(n)(a_{t,j}(n)) denote a
number of partitions (t-cores) of n with BG-rank=j. Here, we provide an elegant
combinatorial proof that 5|p_j(5n+4) by showing that the residue of the 5-core
crank mod 5 divides the partitions enumerated by p_j(5n+4) into five equal
classes. This proof uses the orbit construction in [2] and new identity for
BG-rank. In addition, we find eta-quotient representation for the generating
functions for coefficients a_{t,floor((t+1)/4)}(n), a_{t,-floor((t-1)/4)}(n)
when t is an odd, positive integer. Finally, we derive explicit formulas for
the coefficients a_{5,j}(n) with j=0,1,-1.Comment: 20 pages. This version has an expanded section 7, where we defined
gbg-rank and stated a number of appealing results. We added a new reference.
This paper will appear in Adv. Appl. Mat
K. Saito's Conjecture for Nonnegative Eta Products and Analogous Results for Other Infinite Products
We prove that the Fourier coefficients of a certain general eta product
considered by K. Saito are nonnegative. The proof is elementary and depends on
a multidimensional theta function identity. The z=1 case is an identity for the
generating function for p-cores due to Klyachko [17] and Garvan, Kim and
Stanton [10]. A number of other infinite products are shown to have nonnegative
coefficients. In the process a new generalization of the quintuple product
identity is derived.Comment: 15 pages; greatly expanded version of the earlier 8 page paper
math.NT/060760