A generalization of a 1998 unimodality conjecture of Reiner and Stanton

An interesting, and still wide open, conjecture of Reiner and Stanton predicts that certain "strange" symmetric differences of $q$-binomial coefficients are always nonnegative and unimodal. We extend their conjecture to a broader, and perhaps more natural, framework, by conjecturing that, for each $k\ge 5$, the polynomials $$f(k,m,b)(q)=\binom{m}{k}_q-q^{\frac{k(m-b)}{2}+b-2k+2}\cdot\binom{b}{k-2}_q$$ are nonnegative and unimodal for all $m\gg_k 0$ and $b\le \frac{km-4k+4}{k-2}$ such that $kb\equiv km$ (mod 2), with the only exception of $b=\frac{km-4k+2}{k-2}$ when this is an integer. Using the KOH theorem, we combinatorially show the case $k=5$. In fact, we completely characterize the nonnegativity and unimodality of $f(k,m,b)$ for $k\le 5$. (This also provides an isolated counterexample to Reiner-Stanton's conjecture when $k=3$.) Further, we prove that, for each $k$ and $m$, it suffices to show our conjecture for the largest $2k-6$ values of $b$.Comment: Final version. To appear in the Journal of Combinatoric

A note on the asymptotics of the number of O-sequences of given length

We look at the number $L(n)$ of $O$-sequences of length $n$. Recall that an $O$-sequence can be defined algebraically as the Hilbert function of a standard graded $k$-algebra, or combinatorially as the $f$-vector of a multicomplex. The sequence $L(n)$ was first investigated in a recent paper by commutative algebraists Enkosky and Stone, inspired by Huneke. In this note, we significantly improve both of their upper and lower bounds, by means of a very short partition-theoretic argument. In particular, it turns out that, for suitable positive constants $c_1$ and $c_2$ and all $n>2$, $$e^{c_1\sqrt{n}}\le L(n)\le e^{c_2\sqrt{n}\log n}.$$ It remains an open problem to determine an exact asymptotic estimate for $L(n)$.Comment: Final version to appear in Discrete Mathematics. 2 page

Unimodality of partitions with distinct parts inside Ferrers shapes

We investigate the rank-generating function F[subscript λ] of the poset of partitions contained inside a given shifted Ferrers shape λ. When λ has four parts, we show that F [subscript λ] is unimodal when λ=〈n, n-1, n-2, n-3〉, for any n≥4, and that unimodality fails for the doubly-indexed, infinite family of partitions of the form λ=〈n, n-t, n-2t, n-3t〉, for any given t≥2 and n large enough with respect to t.When λ has b≤3 parts, we show that our rank-generating functions F[subscript λ] are all unimodal. However, the situation remains mostly obscure for b≥5. In general, the type of results that we obtain present some remarkable similarities with those of the 1990 paper of D. Stanton, who considered the case of partitions inside ordinary (straight) Ferrers shapes. Along the way, we also determine some interesting q-analogs of the binomial coefficients, which in certain instances we conjecture to be unimodal. We state several other conjectures throughout this note, in the hopes to stimulate further work in this area. In particular, one of these will attempt to place into a much broader context the unimodality of the posets M(n) of staircase partitions, for which determining a combinatorial proof remains an outstanding open problem.National Science Foundation (U.S.) (Grant DMS-1068625

The Catalan case of Armstrong's conjecture on simultaneous core partitions

A beautiful recent conjecture of D. Armstrong predicts the average size of a partition that is simultaneously an $s$-core and a $t$-core, where $s$ and $t$ are coprime. Our goal is to prove this conjecture when $t=s+1$. These simultaneous $(s,s+1)$-core partitions, which are enumerated by Catalan numbers, have average size $\binom{s+1}{3}/2$.Comment: Some changes in response to the referee's comments. To appear in the SIAM J. on Discrete Mat

On the rank function of a differential poset

We study $r$-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite projective planes, which are shown to be naturally embedded in the initial ranks of some differential poset. As a byproduct, we prove the existence, if and only if $r\geq 6$, of $r$-differential posets nonisomorphic in any two consecutive ranks but having the same rank function. We also show that the Interval Property, conjectured by the second author and collaborators for several sequences of interest in combinatorics and combinatorial algebra, in general fails for differential posets. In the second part, we prove that the rank function $p_n$ of any arbitrary $r$-differential poset has nonpolynomial growth; namely, $p_n\gg n^ae^{2\sqrt{rn}},$ a bound very close to the Hardy-Ramanujan asymptotic formula that holds in the special case of Young's lattice. We conclude by posing several open questions.Comment: A few minor revisions/updates. Published in the Electron. J. Combin. (vol. 19, issue 2, 2012

Emotional crisis in a naturalistic context: characterizing outpatient profiles and treatment effectiveness.

Crisis happens daily yet its understanding is often limited, even in the field of psychiatry. Indeed, a challenge is to assess the potential for change of patients so as to offer appropriate therapeutic interventions and enhance treatment program efficacy. This naturalistic study aimed to identify the socio-demographical characteristics and clinical profiles at admission of patients referred to a specialized Crisis Intervention Center (CIC) and to examine the effectiveness of the intervention. The sample was composed of 352 adult outpatients recruited among the referrals to the CIC. Assessment completed at admission and at discharge examined psychiatric symptoms, defense mechanisms, recovery styles and global functioning. The crisis intervention consisted in a psychodynamically oriented multimodal approach associated with medication. Regarding the clinical profiles at intake, patients were middle-aged (M = 38.56, SD = 10.91), with a higher proportion of women (62.22%). They were addressed to the CIC because they had attempted to commit suicide or had suicidal ideation or presented depressed mood related to interpersonal difficulties. No statistical differences were found between patients dropping out (n = 215) and those attending the crisis intervention (n = 137). Crisis intervention demonstrated a beneficial effect (p < 0.01) on almost all variables, with Effect Sizes (ES) ranging from small to large (0.12 < ES < 0.75; median = 0.49). However, the Reliable Change Index indicated that most of the issues fall into the undetermined category (range 41.46 to 96.35%; median = 66.20%). This study establishes the profile of patients referred to the CIC and shows that more than half of the patients dropped out from the crisis intervention before completion. Our findings suggest that people presenting an emotional crisis benefit from crisis intervention. However, given methodological constraints, these results need to be considered with caution. Moreover, the clinical significance of the improvements is not confirmed. Thus, the effectiveness of crisis intervention in naturalistic context is not fully determined and should be more rigorously studied in future research

Carboxyl-modified single-wall carbon nanotubes improve bone tissue formation in vitro and repair in an in vivo rat model.

The clinical management of bone defects caused by trauma or nonunion fractures remains a challenge in orthopedic practice due to the poor integration and biocompatibility properties of the scaffold or implant material. In the current work, the osteogenic properties of carboxyl-modified single-walled carbon nanotubes (COOH-SWCNTs) were investigated in vivo and in vitro. When human preosteoblasts and murine embryonic stem cells were cultured on coverslips sprayed with COOH-SWCNTs, accelerated osteogenic differentiation was manifested by increased expression of classical bone marker genes and an increase in the secretion of osteocalcin, in addition to prior mineralization of the extracellular matrix. These results predicated COOH-SWCNTs' use to further promote osteogenic differentiation in vivo. In contrast, both cell lines had difficulties adhering to multi-walled carbon nanotube-based scaffolds, as shown by scanning electron microscopy. While a suspension of SWCNTs caused cytotoxicity in both cell lines at levels >20 μg/mL, these levels were never achieved by release from sprayed SWCNTs, warranting the approach taken. In vivo, human allografts formed by the combination of demineralized bone matrix or cartilage particles with SWCNTs were implanted into nude rats, and ectopic bone formation was analyzed. Histological analysis of both types of implants showed high permeability and pore connectivity of the carbon nanotube-soaked implants. Numerous vascularization channels appeared in the formed tissue, additional progenitor cells were recruited, and areas of de novo ossification were found 4 weeks post-implantation. Induction of the expression of bone-related genes and the presence of secreted osteopontin protein were also confirmed by quantitative polymerase chain reaction analysis and immunofluorescence, respectively. In summary, these results are in line with prior contributions that highlight the suitability of SWCNTs as scaffolds with high bone-inducing capabilities both in vitro and in vivo, confirming them as alternatives to current bone-repair therapies

Proof of the Gorenstein Interval Conjecture in low socle degree

Roughly ten years ago, the following "Gorenstein Interval Conjecture" (GIC) was proposed: Whenever $(1,h_1,\dots,h_i,\dots,h_{e-i},\dots,h_{e-1},1)$ and $(1,h_1,\dots,h_i+\alpha,\dots,h_{e-i}+\alpha,\dots,h_{e-1},1)$ are both Gorenstein Hilbert functions for some $\alpha \geq 2$, then $(1,h_1,\dots,h_i+\beta,\dots,h_{e-i}+\beta,\dots,h_{e-1},1)$ is also Gorenstein, for all $\beta =1,2,\dots,\alpha -1$. Since an explicit characterization of which Hilbert functions are Gorenstein is widely believed to be hopeless, the GIC, if true, would at least provide the existence of a strong, and very natural, structural property for such basic functions in commutative algebra. Before now, very little progress was made on the GIC. The main goal of this note is to prove the case $e\le 5$, in arbitrary codimension. Our arguments will be in part constructive, and will combine several different tools of commutative algebra and classical algebraic geometry.Comment: 8 pages, final version. To appear in the J. of Algebr

Spaceflight Effects and Molecular Responses in the Mouse Eye: Observations After Shuttle Mission STS-133

Microgravity-induced cephalad fluid shift and radiation exposure are some of the stressors seen in space exploration. Ocular changes leading to visual impairment in astronauts are of occupational health relevance. Therefore, we analyzed the effects of space flight in the eyes of mice. Six mice were assigned to Flight (FLT), Animal enclosure Module (AEM), or vivarium (VIV) group, respectively. Mice were sacrificed at 1, 5 or 7 days after landing from space. One eye was used for histological and immunohistoche-mistry analysis and the other eye for gene expression profiling. 8-OHdG and caspase-3 immunoreactivity were increased in the retina in FLT samples at return(R+1) compared to AEM/VIV groups, and decreased at day 7 (R+7). beta-amyloid was seen in the nerve fibers at the post-laminar region of the optic nerve in the flight samples (R+7). In addition, oxidative and cellular stress response genes were upregulated in the retina of FLT samples upon landing, and decreased by R+7. According to the results, a reversible molecular damage may occur in the retina of mice exposed to spaceflight followed by protective cellular response

A peptide found in human serum, derived from the c-terminus of albumin, shows antifungal activity in vitro and in vivo

The growing problem of antimicrobial resistance highlights the need for alternative strategies to combat infections. From this perspective, there is a considerable interest in natural molecules obtained from different sources, which are shown to be active against microorganisms, either alone or in association with conventional drugs. In this paper, peptides with the same sequence of fragments, found in human serum, derived from physiological proteins, were evaluated for their antifungal activity. A 13-residue peptide, representing the 597–609 fragment within the albumin C-terminus, was proved to exert a fungicidal activity in vitro against pathogenic yeasts and a therapeutic effect in vivo in the experimental model of candidal infection in Galleria mellonella. Studies by confocal microscopy and transmission and scanning electron microscopy demonstrated that the peptide penetrates and accumulates in Candida albicans cells, causing gross morphological alterations in cellular structure. These findings add albumin to the group of proteins, which already includes hemoglobin and antibodies, that could give rise to cryptic antimicrobial fragments, and could suggest their role in anti-infective homeostasis. The study of bioactive fragments from serum proteins could open interesting perspectives for the development of new antimicrobial molecules derived by natural sources