New Identities for 7-cores with prescribed BG-rank

A q-series with nonnegative power series coefficients is called positive. The partition statistics BG-rank is defined as an alternating sum of parities of parts of a partition. It is known that the generating function for the number of partitions of n that are 7-cores with given BG-rank can be written as certain sum of multi-theta functions. We give explicit representations for these generating functions in terms of sums of positive eta-quotients and derive inequalities for the their coefficients. New identities for the generating function of unrestricted 7-cores and inequalities for their coefficients are also obtained. Our proofs utilize Ramanujan's theory of modular equations.Comment: 12 page

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

On the representations of integers by the sextenary quadratic form x^2+y^2+z^2+ 7s^2+7t^2+ 7u^2 and 7-cores

In this paper we derive an explicit formula for the number of representations of an integer by the sextenary form x^2+y^2+z^2+ 7s^2+7t^2+ 7u^2. We establish the following intriguing inequalities 2b(n)>=a_7(n)>=b(n) for n not equal to 0,2,6,16. Here a_7(n) is the number of partitions of n that are 7-cores and b(n) is the number of representations of n+2 by the sextenary form (x ^2+ y ^2+z ^2+ 7s ^2 + 7t ^2+ 7u^2)/8 with x,y,z,s,t and u being odd.Comment: 10 page

Ramanujan type congruences for quotients of level 7 Klein forms

Klein forms are used to construct generators for the graded algebra of modular forms of level 7. Dissection formulas for the series imply Ramanujan type congruences modulo powers of 7 for a family of generating functions that subsume the counting function for 7-core partitions. The broad class of arithmetic functions considered here enumerate colored partitions by weights determined by parts modulo 7. The method is a prototype for similar analysis of modular forms of level 7 and at other prime levels. As an example of the utility of the dissection method, the paper concludes with a derivation of novel congruences for the number of representations by x^2+xy+2y^2 in exactly k ways

The number of self-conjugate core partitions

A conjecture on the monotonicity of t-core partitions in an article of Stanton [Open positivity conjectures for integer partitions, Trends Math., 2:19-25, 1999] has been the catalyst for much recent research on t-core partitions. We conjecture Stanton-like monotonicity results comparing self-conjugate (t+2)- and t-core partitions of n. We obtain partial results toward these conjectures for values of t that are large with respect to n, and an application to the block theory of the symmetric and alternating groups. To this end we prove formulas for the number of self-conjugate t-core partitions of n as a function of the number of self-conjugate partitions of smaller n. Additionally, we discuss the positivity of self-conjugate 6-core partitions and introduce areas for future research in representation theory, asymptotic analysis, unimodality, and numerical identities and inequalities.Comment: 17 pages, 2 figures, to appear in Journal of Number Theory Updated to accepted version. Removed previously known results about self-conjugate 7-cores and corrected other historical informatio

Soil Microbial Community Dynamics in Response to Prescribed Extreme Fires Following \u3ci\u3eJuniperus virginiana\u3c/i\u3e Invasion in the Loess Canyons of Nebraska

In Nebraska and other regions of the Great Plains, the conifer Juniperus virginiana (eastern redcedar) is converting grasslands to dense woodlands. This is driven by the interacting drivers of fire suppression, altered grazing regimes, climate change and other anthropogenic factors, impacting the provisioning of ecosystem services. This vegetation state transition modifies water resource regulation and biogeochemical cycles leading to altered edaphic properties including soil microbial community composition. To restore these grasslands and control J. virginiana spread, prescribed extreme burns are implemented as a management tool through local prescribed burn associations. We hypothesized that the alternative state transition to dense J. virginiana woodlands leads to a corresponding state transition below-ground that persists post-extreme burn and may facilitate J. virginiana re-establishment. To address this hypothesis, paired grasslands and J. virginiana woodlands in the Loess Canyons of Central Nebraska were subjected to one prescribed extreme burn between 2005 and 2019 to provide a natural burn chronosequence. We quantified J. virginiana re-establishment, soil chemistry, soil microbial biomass and microbial community composition in these paired sites across the chronosequence. Our results partially supported our hypothesis where differences in edaphic variables between J. virginiana sites and grassland sites observed post-burn were largely temporary; however, differences in soil magnesium and microbial community composition were more persistent (\u3e 14 years post-burn). Soil magnesium values were significantly higher in the J. virginiana sites both pre-burn and post-burn across the 14 year chronosequence. Microbial communities were also distinct between J. virginiana and grassland sites pre-burn and across the burn chronosequence. Rapid recovery and/or persistence of specific edaphic factors and soil microbial communities in J. virginiana woodlands post-burn may facilitate early J. virginiana re-establishment. Restoration of historical fire intervals is needed to prevent long term changes to soil function that may facilitate J. virginiana re-establishment. Advisors: Tala Awada and Rhae A. Drijbe