39 research outputs found
On counting permutations by pairs of congruence classes of major index
For a fixed positive integer n, let S_n denote the symmetric group of n!
permutations on n symbols, and let maj(sigma) denote the major index of a
permutation sigma. For positive integers k<m not greater than n and
non-negative integers i and j, we give enumerative formulas for the cardinality
of the set of permutations sigma in S_n with maj(sigma) congruent to i mod k
and maj(sigma^(-1)) congruent to j mod m. When m divides n-1 and k divides n,
we show that for all i,j, this cardinality equals (n!)/(km).Comment: 8 page
Bimahonian distributions
Motivated by permutation statistics, we define for any complex reflection
group W a family of bivariate generating functions. They are defined either in
terms of Hilbert series for W-invariant polynomials when W acts diagonally on
two sets of variables, or equivalently, as sums involving the fake degrees of
irreducible representations for W. It is also shown that they satisfy a
``bicyclic sieving phenomenon'', which combinatorially interprets their values
when the two variables are set equal to certain roots of unity.Comment: Final version to appear in J. London Math. So
A Practical Cryopreservation and Staining Protocol for Immunophenotyping in Population Studies
Large population‐based cohort studies, through their prospective collection of a broad range of health information, represent an invaluable resource for novel insights into the pathogenesis of human diseases. Collection and cryopreservation of viable cells from blood samples is becoming increasingly common in large cohorts as these cells are a valuable resource for immunophenotyping and functional studies. The cryopreservation of peripheral blood mononuclear cells (PBMCs), thawing, and immunophenotyping protocols used to immunophenotype 9938 participants in the Health and Retirement Study (HRS) are described. The extensive quality control involved in a large‐scale immunophenotyping epidemiological study is also outlined. The existing literature on the effect of cryopreservation on various immune cell subsets including T, B, NK cells, monocytes, and dendritic cells is provided. © 2018 by John Wiley & Sons, Inc.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143631/1/cpcy35.pd
On the vanishing of discrete singular cubical homology for graphs
We prove that if G is a graph without 3-cycles and 4-cycles, then the
discrete cubical homology of G is trivial in dimension d, for all d\ge 2. We
also construct a sequence { G_d } of graphs such that this homology is
non-trivial in dimension d for d\ge 1. Finally, we show that the discrete
cubical homology induced by certain coverings of G equals the ordinary singular
homology of a 2-dimensional cell complex built from G, although in general it
differs from the discrete cubical homology of the graph as a whole.Comment: Minor changes, background information adde
Discrete Cubical and Path Homologies of Graphs
In this paper we study and compare two homology theories for (simple and
undirected) graphs. The first, which was developed by Barcelo, Caprano, and
White, is based on graph maps from hypercubes to the graph. The second theory
was developed by Grigor'yan, Lin, Muranov, and Yau, and is based on paths in
the graph. Results in both settings imply that the respective homology groups
are isomorphic in homological dimension one. We show that, for several infinite
classes of graphs, the two theories lead to isomorphic homology groups in all
dimensions. However, we provide an example for which the homology groups of the
two theories are not isomorphic at least in dimensions two and three. We
establish a natural map from the cubical to the path homology groups which is
an isomorphism in dimension one and surjective in dimension two. Again our
example shows that in general the map is not surjective in dimension three and
not injective in dimension two. In the process we develop tools to compute the
homology groups for both theories in all dimensions
Mitochondrial DNA Copy Number Is Associated with Breast Cancer Risk
Mitochondrial DNA (mtDNA) copy number in peripheral blood is associated with increased risk of several cancers. However, data from prospective studies on mtDNA copy number and breast cancer risk are lacking. We evaluated the association between mtDNA copy number in peripheral blood and breast cancer risk in a nested case-control study of 183 breast cancer cases with pre-diagnostic blood samples and 529 individually matched controls among participants of the Singapore Chinese Health Study. The mtDNA copy number was measured using real time PCR. Conditional logistic regression analyses showed that there was an overall positive association between mtDNA copy number and breast cancer risk (Ptrend = 0.01). The elevated risk for higher mtDNA copy numbers was primarily seen for women with <3 years between blood draw and cancer diagnosis; ORs (95% CIs) for 2nd, 3rd, 4th, and 5th quintile of mtDNA copy number were 1.52 (0.61, 3.82), 2.52 (1.03, 6.12), 3.12 (1.31, 7.43), and 3.06 (1.25, 7.47), respectively, compared with the 1st quintile (Ptrend = 0.004). There was no association between mtDNA copy number and breast cancer risk among women who donated a blood sample ≥3 years before breast cancer diagnosis (Ptrend = 0.41). This study supports a prospective association between increased mtDNA copy number and breast cancer risk that is dependent on the time interval between blood collection and breast cancer diagnosis. Future studies are warranted to confirm these findings and to elucidate the biological role of mtDNA copy number in breast cancer risk. © 2013 Thyagarajan et al