A note on the maximum diversity of intersecting families in the symmetric group

Growing evidence indicates that modulation of the catalyst surface microenvironment provides an effective route to optimize electrocatalytic performance, serving as a valuable complement to the traditional emphasis on electronic effects. Yet, despite recent progress in probing interfacial water activation under low-proton conditions, our understanding of this process remains far from complete. Herein, we selected 2,6-diacetylpyridine (DAcPy), a representative molecule with pronounced electrocatalytic enhancement effects, to systematically elucidate the mechanism of interfacial water activation. DAcPy universally enhances reaction kinetics across diverse hydrogen electrocatalytic systems on both Pt and Cu surfaces. Combined scanning tunneling microscope (STM) imaging and theoretical calculations reveal that the symmetric diacetyl groups of DAcPy form a geometrically matched V-shaped molecular vise. This bidentate C═O···H-O hydrogen bonding configuration precisely captures water molecules, and the resulting cooperative polarization effect weakens the H-OH bond, increasing the interfacial water dissociation constant (Kw) by a factor of 2. In situ attenuated total reflection-surface-enhanced infrared absorption spectroscopy (ATR-SEIRAS) measurements directly confirm the formation of a strengthened hydrogen-bond network upon DAcPy modification, in agreement with theoretical predictions. Electrochemical impedance spectroscopy with distribution of relaxation times (EIS-DRT) analysis further reveals that this molecular strategy selectively accelerates the water-involved Volmer and Heyrovsky steps without affecting the Tafel step. Notably, the DAcPy-enabled enhancement applies broadly, significantly boosting rates across various low-proton electrocatalytic systems, including alkaline HER/HOR, CO/CO2 electroreduction to methane/ethylene, and selective acetylene hydrogenation.

It is known that the number of permutations in the symmetric group S 2 n with cycles of odd lengths only is equal to the number of permutations with cycles of even lengths only. We prove a refinement of this equality, involving descent sets: the number of permutations in S 2 n with a prescribed descent set and all cycles of odd lengths is equal to the number of permutations with the complementary descent set and all cycles of even lengths. There is also a variant for S 2 n + 1 . The proof uses generating functions for character values and applies a new identity on higher Lie characters.

In a 2018 paper, Davis and Sagan studied several pattern-avoiding polytopes. They found that a particular pattern-avoiding Birkhoff polytope had the same normalized volume as the order polytope of a certain poset, leading them to ask if the two polytopes were unimodularly equivalent. Motivated by Davis and Sagan’s question, in this paper we define a pattern-avoiding Birkhoff polytope called a c -Birkhoff polytope for each Coxeter element c of the symmetric group. We then show that the c -Birkhoff polytope is unimodularly equivalent to the order polytope of the heap poset of the c -sorting word of the longest permutation. When c = s 1 s 2 ⋯ s n , this result recovers an affirmative answer to Davis and Sagan’s question. Another consequence of this result is that the normalized volume of the c -Birkhoff polytope is the number of the longest chains in the (type A) c -Cambrian lattice.

On the number and sizes of double cosets of Sylow subgroups of the symmetric group

The Bonferroni mean operator, as a powerful aggregation operator, is widely applied as a solution to various problems, owing to its strong ability to capture relevance between different variables. Compared with other extensions, the weighted hesitant fuzzy set (WHFS) can depict the fuzziness of relationships between things better. Considering the advantage of weighted hesitant fuzzy sets (describing fuzziness more objectively in real problems), the Bonferroni mean is introduced into weighted hesitant fuzzy set theory. In this paper, by merging and transforming weighted hesitant fuzzy weighted average/geometric (WHFWA/WHFWG) operators, a novel weighted hesitant fuzzy geometric Bonferroni mean (WHFGBM) operator is developed using weighted hesitant fuzzy theory, so as to fuse information and provide idea support for practical tasks under various experts’ common judgment. To show the effectiveness of the novel operator intuitively, the comparison results of symmetric numeric examples are displayed.

Interlimb asymmetries may influence contralateral knee osteoarthritis (OA) progression, yet research remains unclear. This study examined whether patient-reported outcomes and knee biomechanics differ between individuals with knee OA exhibiting symmetrical versus asymmetrical knee loading. Forty-three individuals with knee OA were dichotomized into symmetrical (≤14% asymmetry; n = 19) and asymmetrical (>14% asymmetry; n = 24) groups based on total joint moment symmetry indices. Participants completed the Knee Injury and Osteoarthritis Outcome Score and Intermittent and Constant Osteoarthritis Pain questionnaires. Three-dimensional kinematics and kinetics were collected during walking at a self-selected speed. Independent t tests and statistical parametric mapping examined between-group differences in patient-reported outcomes and biomechanical measures. Individuals with symmetrical knee loading had worse Knee Injury and Osteoarthritis Outcome Score activities of daily living scores (P = .041) than those with asymmetrical loading. Individuals with symmetrical knee loading exhibited less knee extension moment during late stance (P = .031) and lower knee adduction moment range in their affected knee compared with asymmetrical loaders. Individuals with symmetrical knee loading walked with lower knee flexion angles (P = .011), less midstance unloading (P = .011), and lower peak knee flexion moment (P < .001) in their contralateral knee compared with asymmetrical loaders. Symmetrical knee loading was associated with affected and contralateral knee biomechanics that were consistent with more severe knee OA and worse functional outcomes.

Abstract Blundell, Buesing, Davies, Veličković, and Williamson (BBDVW) introduced the notion of a hypercube decomposition of an interval in Bruhat order. They conjectured a recursive formula in terms of this structure that, if shown for all intervals, would imply the Combinatorial Invariance Conjecture of Lusztig and Dyer, for Kazhdan–Lusztig polynomials of the symmetric group. In this article, we prove implications between the BBDVW Conjecture and several other recurrences for hypercube decompositions, under varying hypotheses, which have appeared in the recent literature. As an application, we prove the BBDVW Conjecture for lower intervals $[e,v]$, the first non-trivial class of intervals for which it has been established.

Abstract Quandles are algebraic structures showing up in different mathematical contexts. A group 𝐺 with the conjugation operation forms a quandle, Conj ⁡ ( G ) \operatorname{Conj}(G) . In the opposite direction, a group As ⁡ ( Q ) \operatorname{As}(Q) can be constructed out of any quandle 𝑄. We explore As ⁡ ( Conj ⁡ ( G ) ) \operatorname{As}(\operatorname{Conj}(G)) for a group 𝐺 admitting a presentation with only conjugation and power relations. Symmetric groups S n S_{n} are typical examples. For such groups, we show that As ⁡ ( Conj ⁡ ( G ) ) \operatorname{As}(\operatorname{Conj}(G)) injects into G × Z m G\times\mathbb{Z}^{m} , where 𝑚 is the number of conjugacy classes of 𝐺. From this, we deduce information about the torsion, centre, and derived group of As ⁡ ( Conj ⁡ ( G ) ) \operatorname{As}(\operatorname{Conj}(G)) . As an application, we compute the second integral quandle homology group of Conj ⁡ ( S n ) \operatorname{Conj}(S_{n}) , and unveil rich torsion therein.

Cerebral malperfusion in acute type A aortic dissection (ATAAD) is a serious condition. Predicting postoperative neurological outcomes is important to decide treatment strategies; however, current prediction methods have limitations. Therefore, this study examined whether the lateral ventricular volume ratio (LVR) on preoperative head computed tomography (CT) can predict postoperative neurological outcomes. Among patients who underwent surgery for ATAAD at our institution between January 2007 and August 2024, those with cerebral malperfusion who underwent preoperative head CT were included. Cerebral malperfusion was defined as common carotid artery true lumen stenosis ≥50% and new neurological symptoms. The LVR was calculated as the larger lateral ventricular volume divided by the smaller lateral ventricular volume. Of 386 patients with ATAAD, 33 had cerebral malperfusion and underwent preoperative CT. Receiver operating characteristic analysis determined LVR < 1.067 as the cut-off, classifying patients into the symmetric group (ratio < 1.067, n = 10) or the asymmetric group (ratio ≥ 1.067, n = 23). The symmetric group showed 80% postoperative neurological recovery compared with 26% in the asymmetric group (P = .007). The postoperative modified Rankin Scale score was ≥4 in 20% and 78% of patients in the symmetric and asymmetric groups, respectively (P = .005). Univariable analysis identified lateral ventricular symmetry as a predictor of postoperative neurological recovery (odds ratio = 11.3, 95% confidence interval: 1.86-69.1, P = .008). Long-term overall survival was significantly better in the symmetric group than in the asymmetric group (P = .043). In patients with ATAAD and cerebral malperfusion, lateral ventricular symmetry on preoperative head CT may predict postoperative neurological outcomes. 4707.

The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups

In this work, we present an algorithmic treatment of the representation theory of the algebra of partially transposed permutation operators, denoted byAp,pd, which is a matrix representation of the abstract walled Brauer algebra. We provide an explicit and fully developed framework for constructing irreducible matrix units within the algebra. In contrast to the established earlier Gelfand-Tsetlin type constructions, the presented matrix units are adapted to the action of the subalgebraC[Sp]×C[Sp], whereSpis the symmetric group. What is more, the basis is constructed in such a way that it produces the decomposition of the algebra into a direct sum of ideals, in contrast to its nested structure considered before. The decomposition of this kind has not been considered before in full generality. Our method reveals a recursive scheme for generating irreducible matrix units in all ideals ofAp,pd, offering a systematic approach that applies to small system sizes and arbitrary local dimensions. We apply the developed formalism to the algebraA2,2dand illustrate the algorithm in practice. In addition, using the constructed basis, we proved a novel contraction theorem for the elements fromA3,3d, which is the starting point for further investigations.

BackgroundRobotic mitral valve repair for mitral regurgitation (MR) is increasingly popular due to shorter hospital stays, reduced perioperative pain, and faster recovery. Barlow’s disease is a significant subset of primary MR due to its complex repair requirements. This study presents our surgical and short-term outcomes for patients with Barlow’s disease treated using the da Vinci Xi robotic system.MethodsSince adopting robotic surgery in June 2021, all consecutive patients diagnosed with Barlow’s disease through September 2023 were included. Based on preoperative echocardiography, patients were classified into two groups: “symmetrical”, involving bileaflet prolapse and a central regurgitant jet, and “asymmetrical”, with single-leaflet prolapse and eccentric regurgitation.ResultsThirty-six patients with Barlow’s disease underwent robotic surgery during the study period. Repair success was 100%, with no conversions required. No significant differences were found in baseline characteristics between the prolapse types. Annuloplasty alone was sufficient for the symmetrical group, whereas asymmetrical cases required neochord loops, reflected by shorter aortic cross-clamp (190 vs. 151 min; P=0.034) and aortic cross-clamp (121 vs. 89 min; P=0.003) durations in the symmetrical group. During a median follow-up of 7.5 months, there were no deaths or reinterventions. Additionally, no cases of progression to ≥ moderate MR were observed during the median echocardiographic follow-up of 5.3 months.ConclusionsRobotic repair of Barlow’s disease is safe, feasible, and yields excellent short-term outcomes. For patients with symmetrical prolapse, annuloplasty alone provides effective repair.

The sudden polarization (SP) effect converts a resonant zwitterionic state into a polarized charge-separated (CS) state, yet its dynamics in stable diradicaloids remain insufficiently understood. Here we investigate a pair of C2-symmetric sulfone-functionalized Chichibabin's hydrocarbons with nearly degenerate zwitterionic states but differing in symmetric electron-withdrawing groups (EWGs). Femtosecond transient absorption spectroscopy directly captures subpicosecond SP processes in both systems and, together with quantum-chemical calculations, reveals the thermodynamic accessibility of the CS state governed by solvent polarity and excitonic coupling. Increasing solvent polarity facilitates the SP effect, which leads to full charge separation via the relaxation of polarized zwitterionic states. Additionally, weakening excitonic coupling through the substituent effect allows the CS state to be accessible across all polar media. These results establish the SP effect as the trigger for symmetry breaking and establish tunable substituents and dielectric environments as effective handles for directing SP relaxation dynamics in symmetric diradicaloids.

Combinatorial transition matrices arise frequently in the theory of symmetric functions and their generalizations. The entries of such matrices often count signed, weighted combinatorial structures such as semistandard tableaux, rim-hook tableaux, or brick tabloids. Bijective proofs that two such matrices are inverses of each other may be difficult to find. This paper presents a general framework for proving such inversion results in the case where the combinatorial objects are built up recursively by successively adding some incremental structure such as a single horizontal strip or rim-hook. In this setting, we show that a sequence of matrix inversion results $A_nB_n=I$ can be reduced to a certain "local" identity involving the incremental structures. Here, $A_n$ and $B_n$ are matrices that might be non-square, and the columns of $A_n$ and the rows of $B_n$ are indexed by compositions of $n$. We illustrate the general theory with four classical applications involving the Kostka matrices, the character tables of the symmetric group, incidence matrices for composition posets, and matrices counting brick tabloids. We obtain a new, canonical bijective proof of an inversion result for rectangular Kostka matrices, which complements the proof for the square case due to Eğecioğlu and Remmel. We also give a new bijective proof of the orthogonality result for the irreducible $S_n$-characters that is shorter than the original version due to White.

The power graph P(G) is a simple graph associated with a group G that represents power relations among its elements. Although power graphs have been widely studied in connection with domination and connectivity, the effect of removing dominating sets, particularly those excluding the identity element, on graph connectivity has not been examined in detail. This study aims to characterize dominating sets in power graphs of finite groups and to investigate whether connectivity is preserved after their removal, with emphasis on symmetric groups and cyclic groups. This research employs a theoretical and analytical approach based on group theory and algebraic graph theory. The results show that, for symmetric groups Sn, there exists a dominating set excluding the identity element such that the power graph remains connected after its removal. Furthermore, for cyclic groups Cn, any generator forms a minimum dominating set, and the power graph remains connected after its removal.

Abstract We study the orders of products of two class transpositions in the group CT ⁡ ( Z ) \operatorname{CT}(Z) , a simple subgroup of the symmetric group on the integers. For pairs of class transpositions sharing a common vertex, we prove that the order of their product is either 1, 3, or ∞, and provide a precise criterion for the infinite order case. Furthermore, we investigate pairs of equal-residue and equal-modulus class transpositions, establishing conditions under which their product has finite or infinite order. Our results provide a partial answer to a question posed in the Kourovka notebook (see Question 18.48).

This note is in two parts. The first part contains proofs of four elementary criteria for two permutations in the symmetric group Sn to be conjugate in S n . Most readers know the first criterion but perhaps fewer are familiar with the remaining three. The second part contains a discussion of the problem of deciding whether two ordered pairs of permutations are conjugate by an element in S n . In sharp contrast to the situation in part one, here no elementary criteria are known. However, pairs of permutations give rise to dessins d’enfants and two ordered pairs of permutations are conjugate precisely when the corresponding dessins are isomorphic. We give an expository discussion, with examples and references but no proofs, of some of this material; we hope our discussion will help readers who are encountering this known material for the first time.

Three indan-2-one-based donor–π–acceptor–π–donor type dyes with symmetric donor groups were synthesized and characterized to study their nonlinear optical (NLO) properties and their potential use in the rapid and selective determination of cyanide. The designed structures feature symmetrical alkylaminophenyl donor groups and a strong electron-withdrawing dicyanovinylene as an acceptor connected through vinyl groups as a π-bridge. These strongly π-conjugated organic dyes can absorb in the NIR region, and they showed sensitivity towards the polarity of solvents with colorimetric and optical changes. Because of the strong donor–acceptor structure, second-order NLO properties were studied by measuring electric field-induced second harmonic (EFISH) values, which showed significant second-order NLO responses. The experimental results were explained using density functional theory (DFT) methods. The dyes also exhibit chemosensor properties, showing selectivity for cyanide via a Michael addition mechanism that causes the disappearance of the ICT band, and a significant color change was observed in both organic and aqueous media. In addition, the interaction mechanism between cyanide and the chemosensor is determined by a 1H NMR study and explained by DFT calculations.

We study the blocks for Ariki–Koike algebras using a general notion of core for [Formula: see text]-partitions. We interpret the action of the affine symmetric group on the blocks in the context of level rank duality and study the orbits under this action.