The signed Eulerian numbers on involutions

We define an analogue of signed Eulerian numbers $f_{n,k}$ for involutions of the symmetric group and derive some combinatorial properties of this sequence. In particular, we exhibit both an explicit formula and a recurrence for $f_{n,k}$ arising from the properties of its generating function.Comment: 10 page

Permutations and Pairs of Dyck Paths

We define a map v between the symmetric group Sn and the set of pairs of Dyck paths of semilength n. We show that the map v is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map

Consecutive patterns in restricted permutations and involutions

It is well-known that the set $\mathbf I_n$ of involutions of the symmetric group $\mathbf S_n$ corresponds bijectively - by the Foata map $F$ - to the set of $n$-permutations that avoid the two vincular patterns $\underline{123},$ $\underline{132}.$ We consider a bijection $\Gamma$ from the set $\mathbf S_n$ to the set of histoires de Laguerre, namely, bicolored Motzkin paths with labelled steps, and study its properties when restricted to $\mathbf S_n(1\underline{23},1\underline{32}).$ In particular, we show that the set $\mathbf S_n(\underline{123},{132})$ of permutations that avoids the consecutive pattern $\underline{123}$ and the classical pattern $132$ corresponds via $\Gamma$ to the set of Motzkin paths, while its image under $F$ is the set of restricted involutions $\mathbf I_n(3412).$ We exploit these results to determine the joint distribution of the statistics des and inv over $\mathbf S_n(\underline{123},{132})$ and over $\mathbf I_n(3412).$ Moreover, we determine the distribution in these two sets of every consecutive pattern of length three. To this aim, we use a modified version of the well-known Goulden-Jacson cluster method.Comment: 24 page

Trisecant Lemma for Non Equidimensional Varieties

The classic trisecant lemma states that if $X$ is an integral curve of \PP^3 then the variety of trisecants has dimension one, unless the curve is planar and has degree at least 3, in which case the variety of trisecants has dimension 2. In this paper, our purpose is first to present another derivation of this result and then to introduce a generalization to non-equidimensional varities. For the sake of clarity, we shall reformulate our first problem as follows. Let $Z$ be an equidimensional variety (maybe singular and/or reducible) of dimension $n$, other than a linear space, embedded into \PP^r, $r \geq n+1$. The variety of trisecant lines of $Z$, say $V_{1,3}(Z)$, has dimension strictly less than $2n$, unless $Z$ is included in a $(n+1)-$dimensional linear space and has degree at least 3, in which case $\dim(V_{1,3}(Z)) = 2n$. Then we inquire the more general case, where $Z$ is not required to be equidimensional. In that case, let $Z$ be a possibly singular variety of dimension $n$, that may be neither irreducible nor equidimensional, embedded into \PP^r, where $r \geq n+1$, and $Y$ a proper subvariety of dimension $k \geq 1$. Consider now $S$ being a component of maximal dimension of the closure of \{l \in \G(1,r) \vtl \exists p \in Y, q_1, q_2 \in Z \backslash Y, q_1,q_2,p \in l\}. We show that $S$ has dimension strictly less than $n+k$, unless the union of lines in $S$ has dimension $n+1$, in which case $dim(S) = n+k$. In the latter case, if the dimension of the space is stricly greater then $n+1$, the union of lines in $S$ cannot cover the whole space. This is the main result of our work. We also introduce some examples showing than our bound is strict

Recognizing menopause in women with amenorrhea induced by cytotoxic chemotherapy for endocrine-responsive early breast cancer.

Abstract Cytotoxic anticancer treatment may induce amenorrhea or menopause to a variable extent. These side effects may not only impair or impede fertility but also cause sexual dysfunction, bone loss, and menopausal symptoms, with a strikingly negative effect on quality of life in many women. Aromatase inhibitors (AIs) are a recommended adjuvant endocrine treatment option in postmenopausal patients affected by early breast cancer (EBC) but are contraindicated in premenopausal women and in those with residual ovarian function. Women over 40 years of age with chemotherapy-induced amenorrhea (CIA) and routine hormonal levels consistent with menopause may receive an AI as adjuvant endocrine treatment. For these women, the tools available to identify menopause do not appear to be completely reliable. This review focused on the pathophysiology of ovarian toxicity induced by cytotoxic agents and on potentially useful methods to diagnose chemotherapy-induced menopause in patients treated with adjuvant chemotherapy for endocrine-responsive EBC. Moreover, practical approaches are proposed to distinguish true menopausal women, who would benefit from AIs, from those with transient or persistent CIA

Specific and non-specific biomarkers in neuroendocrine gastroenteropancreatic tumors

The diagnosis of neuroendocrine tumors (NETs) is a challenging task: Symptoms are rarely specific, and clinical manifestations are often evident only when metastases are already present. However, several bioactive substances secreted by NETs can be included for diagnostic, prognostic, and predictive purposes. Expression of these substances differs between different NETs according to the tumor hormone production. Gastroenteropancreatic (GEP) NETs originate from the diffuse neuroendocrine system of the gastrointestinal tract and pancreatic islets cells: These tumors may produce many non-specific and specific substances, such as chromogranin A, insulin, gastrin, glucagon, and serotonin, which shape the clinical manifestations of the NETs. To provide an up-to-date reference concerning the different biomarkers, as well as their main limitations, we reviewed and summarized existing literature

A rare case of pituitary melanoma metastasis: a dramatic and prolonged response to dabrafenib-trametinib therapy

Introduction: Pituitary metastases (PM) are rare events and to date only very few cases of melanoma PM have been described in literature up to now. Case Presentation: We describe the clinical history of a 33-year-old male patient who underwent surgical excision of an inter-scapular melanoma in 2008. The subsequent follow-up was negative for ~10 years. In September 2018, due to the onset of a severe headache, the patient underwent a brain magnetic resonance imaging, which showed an expansive mass in the saddle and suprasellar region with a maximum diameter of 17 mm. Pituitary function tests and visual field were normal. Worsening of the headache and the appearance of a left eye ptosis led the patient to surgical removal of the lesion in October 2018. The histological examination unexpectedly showed metastasis of the melanoma. Post-operative hormonal assessment showed secondary hypothyroidism and hypoadrenalism, which were both promptly treated, and a mild hypogonadism. Three months after surgery, a sellar MRI showed a persistent, increased pituitary mass (3 cm of diameter); fluorine-18-fluorodeoxyglucose positron emission tomography/computed tomography (18F-FDG PET/CT) detected an increased radiopharmaceutical uptake in the sellar region. Due to the persistence of the disease and the evidence of a BRAF V600E mutation, in February 2019, the patient underwent a combined treatment with dabrafenib (a BRAF inhibitor) and trametinib (mitogen-activated extracellular signal-regulate kinase inhibitor). Sellar MRI performed 6 months later showed no evidence of mass in the sellar region. The patient was in a good clinical condition and did not complain of headaches or other symptoms; there were no significant side-effects from the anticancer therapy. After 13 months of treatment, the patient showed no recurrence of the disease on morphological imaging. Anticancer therapy was confirmed, replacement therapies with hydrocortisone and levothyroxine continued and the pituitary-gonadal axis was restored. Conclusion: This is a very interesting case, both for the rarity of the pituitary melanoma metastasis and for the singular therapeutic course carried out by the patient. This is the first case of a pituitary melanoma metastasis with BRAF mutation, successfully treated with the combination of dabrafenib and trametinib after incomplete surgical removal