Proof of a conjectured q,t-Schr\"{o}der identity

A conjecture of Chunwei Song on a limiting case of the q,t-Schr\"{o}der theorem is proved combinatorially. The proof matches pairs of tableaux to Catalan words in a manner that preserves differences in the maj statistic.Comment: 8 pages; v2 corrects presentation error in example and notation (substance of proof unchanged

Distribution of Fishes in Reference Streams Within Arkansas\u27 Ecoregions

The State of Arkansas has been subdivided into six ecoregions based on the homogeneity of land surface forms, potential natural vegetation, soil types and land uses. Reference streams of various sizes, excluding the large rivers, and with the least amount of point source and non-point source disturbances were selected for intensive physical, chemical and biological sampling. These data are to be used to characterize the streams and establish water quality criteria which will protect all stream uses. Fish communities of the reference streams were distinctively different among the ecoregions and can easily be used to characterize the waters of different ecoregions. Although composed of different species, the composition of trophic feeding levels of the fish community was very similar among the ecoregions. The average number of species collected per sample site was similar among the ecoregions; however, the Arkansas River Valley and the Gulf Coastal ecoregions had the greatest species richness and the Delta ecoregion was the lowest in species richness. Species of fish sensitive to environmental change comprised near 50% or more of the community relative abundance in the Boston Mountains, Ozark Highlands and Ouachita Mountains ecoregions. Delta ecoregion fish populations contained less than 1% sensitive species. Comparisons of the ten most abundant species from each ecoregion by use of a similarity index shows very little similarity among the ecoregions. The Ouachita Mountains and Boston Mountains communities were most similar and the Ozark Highlands community versus Delta and Ozark Highlands versus Gulf Coastal were least similar

The part-frequency matrices of a partition

A new combinatorial object is introduced, the part-frequency matrix sequence of a partition, which is elementary to describe and is naturally motivated by Glaisher's bijection. We prove results that suggest surprising usefulness for such a simple tool, including the existence of a related statistic that realizes every possible Ramanujan-type congruence for the partition function. To further exhibit its research utility, we give an easy generalization of a theorem of Andrews, Dixit and Yee on the mock theta functions. Throughout, we state a number of observations and questions that can motivate an array of investigations.Comment: Presented at the Kliakhandler Conference 2015, Algebraic Combinatorics and Applications, at Michigan Technological University. To appear in the Proceeding

Partitions into a small number of part sizes

We study $\nu_k(n)$, the number of partitions of $n$ into $k$ part sizes, and find numerous arithmetic progressions where $\nu_2$ and $\nu_3$ take on values divisible by 2 and 4. Expanding earlier work, we show $\nu_2(An+B) \equiv 0 \pmod{4}$ for (A,B) = (36,30), (72,42), (252,114), (196,70), and likely many other progressions for which our method should easily generalize. Of some independent interest, we prove that the overpartition function $\bar{p}(n) \equiv 0 \pmod{16}$ in the first three progressions (the fourth is known), and thereby show that $\nu_3(An+B) \equiv 0 \pmod{2}$ in each of these progressions as well, and discuss the relationship between these congruences in more generality. We end with open questions in this area.Comment: 11 pages; v2, small correction to proof of Theorem 7; v3, clean up some explanations, acknowledge recent results from Xinhua Xiong on overpartitions mod 16; v4, final journal version to appear International Journal of Number Theory (Feb. 2017

A Bijection for Partitions with Initial Repetitions

A theorem of Andrews equates partitions in which no part is repeated more than 2k-1 times to partitions in which, if j appears at least k times, all parts less than j also do so. This paper proves the theorem bijectively, with some of the generalizations that usually arise from such proofs.Comment: 5 page

The effect of land disposal of dairy factory wastes on soil properties : a thesis presented in partial fulfilment of the requirements for the degree of Master of Agricultural Science in Soil Science, Massey University

Many New Zealand dairy factories dispose of their wastewater by spray irrigating onto pasture. Little is known, however, about the effects of this disposal on soil properties. Research was undertaken at three pasture disposal sites in order to determine whether certain soil property changes may have occurred as a result of the wastewater treatment. Of particular interest were those properties related to water movement. Laboratory studies using 'undisturbed' soil cores indicated that dairy factory wastewater can impede soil water movement. A single application of simulated whey effluent resulted in approximately a 50% decrease in saturated hydraulic conductivity (K) within two days. This reduction was observed to be caused by a combination of both physical and biological blockage processes. With repetitive doses of effluent a K decrease of over 99% was induced in some cores. Several cores, particularly those containing earthworms, showed signs of recovery, and in some cores the final hydraulic conductivity value was greater than the initial value. Analyses of soil samples from the disposal and control sites at Te Rehunga and Tokomaru suggest that fifteen years of wastewater irrigation have resulted in marked changes in soil physical, chemical and biological properties. Total carbon and nitrogen levels were found to be significantly higher at the disposal site; for the Te Rehunga site, the differences in the organic matter level down to 600mm represented an increase of 250 000 kg ha-1. Water balances for the Te Rehunga and Longburn sites indicate that, in the absence of wastewater, pasture is likely to be water stressed on average for approximately forty days per year. The water balance also shows that deep percolation will be greatly increased by the wastewater application. The period of maximum deep percolation loss is likely to be September to October at both the Te Rehunga and Longburn disposal sites. The major site management problems encountered at the disposal sites examined occurred as a result of poor soil drainage, pasture burning and pasture pulling. An infiltration problem was observed at the Longburn site and the recently established disposal site at Tokomaru, with two major causes of the low infiltration rate appearing to be blockage from the effluent and pugging; these observations illustrate the need for controlling the effluent application rate, the suspended solids level in the wastewater, and the stock grazing pattern, in order to minimise site drainage problems. A drainage problem over the winter-spring period at Te Rehunga was due to a high groundwater table. Pasture burning was observed at all three disposal sites. The pasture pulling problem at Te Rehunga is the only cited example of such a problem occurring at a dairy factory disposal site. Observations made at the established Te Rehunga and Tokomaru disposal sites show that long term spray irrigation of dairy factory wastewater can occur without inducing undesirable soil property changes. It appears as though considerable benefit can be gained from the wastewater irrigation, particularly in reducing the incidence of water stress in the pasture and decreasing the requirement for fertilizer

Congruences for 9-regular partitions modulo 3

It is proved that the number of 9-regular partitions of n is divisible by 3 when n is congruent to 3 mod 4, and by 6 when n is congruent to 13 mod 16. An infinite family of congruences mod 3 holds in other progressions modulo powers of 4 and 5. A collection of conjectures includes two congruences modulo higher powers of 2 and a large family of "congruences with exceptions" for these and other regular partitions mod 3.Comment: 7 pages. v2: added citations and proof of one conjecture from a reader. Submitted versio