skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.


Title: Cyclic Shuffle-Compatibility Via Cyclic Shuffle Algebras
Abstract A permutation statistic$${{\,\textrm{st}\,}}$$ st is said to be shuffle-compatible if the distribution of$${{\,\textrm{st}\,}}$$ st over the set of shuffles of two disjoint permutations$$\pi $$ π and$$\sigma $$ σ depends only on$${{\,\textrm{st}\,}}\pi $$ st π ,$${{\,\textrm{st}\,}}\sigma $$ st σ , and the lengths of$$\pi $$ π and$$\sigma $$ σ . Shuffle-compatibility is implicit in Stanley’s early work onP-partitions, and was first explicitly studied by Gessel and Zhuang, who developed an algebraic framework for shuffle-compatibility centered around their notion of the shuffle algebra of a shuffle-compatible statistic. For a family of statistics called descent statistics, these shuffle algebras are isomorphic to quotients of the algebra of quasisymmetric functions. Recently, Domagalski, Liang, Minnich, Sagan, Schmidt, and Sietsema defined a version of shuffle-compatibility for statistics on cyclic permutations, and studied cyclic shuffle-compatibility through purely combinatorial means. In this paper, we define the cyclic shuffle algebra of a cyclic shuffle-compatible statistic, and develop an algebraic framework for cyclic shuffle-compatibility in which the role of quasisymmetric functions is replaced by the cyclic quasisymmetric functions recently introduced by Adin, Gessel, Reiner, and Roichman. We use our theory to provide explicit descriptions for the cyclic shuffle algebras of various cyclic permutation statistics, which in turn gives algebraic proofs for their cyclic shuffle-compatibility.  more » « less
Award ID(s):
2316181
PAR ID:
10470568
Author(s) / Creator(s):
; ;
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Annals of Combinatorics
Volume:
28
Issue:
2
ISSN:
0218-0006
Format(s):
Medium: X Size: p. 615-654
Size(s):
p. 615-654
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract The elliptic flow$$(v_2)$$ ( v 2 ) of$${\textrm{D}}^{0}$$ D 0 mesons from beauty-hadron decays (non-prompt$${\textrm{D}}^{0})$$ D 0 ) was measured in midcentral (30–50%) Pb–Pb collisions at a centre-of-mass energy per nucleon pair$$\sqrt{s_{\textrm{NN}}} = 5.02$$ s NN = 5.02  TeV with the ALICE detector at the LHC. The$${\textrm{D}}^{0}$$ D 0 mesons were reconstructed at midrapidity$$(|y|<0.8)$$ ( | y | < 0.8 ) from their hadronic decay$$\mathrm {D^0 \rightarrow K^-\uppi ^+}$$ D 0 K - π + , in the transverse momentum interval$$2< p_{\textrm{T}} < 12$$ 2 < p T < 12  GeV/c. The result indicates a positive$$v_2$$ v 2 for non-prompt$${{\textrm{D}}^{0}}$$ D 0 mesons with a significance of 2.7$$\sigma $$ σ . The non-prompt$${{\textrm{D}}^{0}}$$ D 0 -meson$$v_2$$ v 2 is lower than that of prompt non-strange D mesons with 3.2$$\sigma $$ σ significance in$$2< p_\textrm{T} < 8~\textrm{GeV}/c$$ 2 < p T < 8 GeV / c , and compatible with the$$v_2$$ v 2 of beauty-decay electrons. Theoretical calculations of beauty-quark transport in a hydrodynamically expanding medium describe the measurement within uncertainties. 
    more » « less
  2. Abstract The total charm-quark production cross section per unit of rapidity$$\textrm{d}\sigma ({{\textrm{c}}\overline{\textrm{c}}})/\textrm{d}y$$ d σ ( c c ¯ ) / d y , and the fragmentation fractions of charm quarks to different charm-hadron species$$f(\textrm{c}\rightarrow {\textrm{h}}_{\textrm{c}})$$ f ( c h c ) , are measured for the first time in p–Pb collisions at$$\sqrt{s_\textrm{NN}} = 5.02~\text {Te}\hspace{-1.00006pt}\textrm{V} $$ s NN = 5.02 Te V at midrapidity ($$-0.96<0.04$$ - 0.96 < y < 0.04 in the centre-of-mass frame) using data collected by ALICE at the CERN LHC. The results are obtained based on all the available measurements of prompt production of ground-state charm-hadron species:$$\textrm{D}^{0}$$ D 0 ,$$\textrm{D}^{+}$$ D + ,$$\textrm{D}_\textrm{s}^{+}$$ D s + , and$$\mathrm {J/\psi }$$ J / ψ mesons, and$$\Lambda _\textrm{c}^{+}$$ Λ c + and$$\Xi _\textrm{c}^{0}$$ Ξ c 0 baryons. The resulting cross section is$$ \textrm{d}\sigma ({{\textrm{c}}\overline{\textrm{c}}})/\textrm{d}y =219.6 \pm 6.3\;(\mathrm {stat.}) {\;}_{-11.8}^{+10.5}\;(\mathrm {syst.}) {\;}_{-2.9}^{+8.3}\;(\mathrm {extr.})\pm 5.4\;(\textrm{BR})\pm 4.6\;(\mathrm {lumi.}) \pm 19.5\;(\text {rapidity shape})+15.0\;(\Omega _\textrm{c}^{0})\;\textrm{mb} $$ d σ ( c c ¯ ) / d y = 219.6 ± 6.3 ( stat . ) - 11.8 + 10.5 ( syst . ) - 2.9 + 8.3 ( extr . ) ± 5.4 ( BR ) ± 4.6 ( lumi . ) ± 19.5 ( rapidity shape ) + 15.0 ( Ω c 0 ) mb , which is consistent with a binary scaling of pQCD calculations from pp collisions. The measured fragmentation fractions are compatible with those measured in pp collisions at$$\sqrt{s} = 5.02$$ s = 5.02 and 13 TeV, showing an increase in the relative production rates of charm baryons with respect to charm mesons in pp and p–Pb collisions compared with$$\mathrm {e^{+}e^{-}}$$ e + e - and$$\mathrm {e^{-}p}$$ e - p collisions. The$$p_\textrm{T}$$ p T -integrated nuclear modification factor of charm quarks,$$R_\textrm{pPb}({\textrm{c}}\overline{\textrm{c}})= 0.91 \pm 0.04\;\mathrm{(stat.)} ^{+0.08}_{-0.09}\;\mathrm{(syst.)} ^{+0.05}_{-0.03}\;\mathrm{(extr.)} \pm 0.03\;\mathrm{(lumi.)}$$ R pPb ( c c ¯ ) = 0.91 ± 0.04 ( stat . ) - 0.09 + 0.08 ( syst . ) - 0.03 + 0.05 ( extr . ) ± 0.03 ( lumi . ) , is found to be consistent with unity and with theoretical predictions including nuclear modifications of the parton distribution functions. 
    more » « less
  3. Abstract The strange quark content of the proton is probed through the measurement of the production cross section for a W boson and a charm (c) quark in proton–proton collisions at a center-of-mass energy of 13$$\,\text {Te}\hspace{-.08em}\text {V}$$ Te V . The analysis uses a data sample corresponding to a total integrated luminosity of 138$$\,\text {fb}^{-1}$$ fb - 1 collected with the CMS detector at the LHC. The W bosons are identified through their leptonic decays to an electron or a muon, and a neutrino. Charm jets are tagged using the presence of a muon or a secondary vertex inside the jet. The$$\hbox {W}+\hbox {c}$$ W + c production cross section and the cross section ratio$$R_\textrm{c}^{\pm }= \sigma ({\hbox {W}}^{+}+\bar{\text {c}})/\sigma (\hbox {W}^{-}+{\textrm{c}})$$ R c ± = σ ( W + + c ¯ ) / σ ( W - + c ) are measured inclusively and differentially as functions of the transverse momentum and the pseudorapidity of the lepton originating from the W boson decay. The precision of the measurements is improved with respect to previous studies, reaching 1% in$$R_\textrm{c}^{\pm }= 0.950 \pm 0.005\,\text {(stat)} \pm 0.010 \,\text {(syst)} $$ R c ± = 0.950 ± 0.005 (stat) ± 0.010 (syst) . The measurements are compared with theoretical predictions up to next-to-next-to-leading order in perturbative quantum chromodynamics. 
    more » « less
  4. Abstract This paper presents the first measurement of$$\psi {(2S)}$$ ψ ( 2 S ) and$$\chi _{c1}(3872)$$ χ c 1 ( 3872 ) meson production within fully reconstructed jets. Each quarkonium state (tag) is reconstructed via its decay to the$${{J \hspace{-1.66656pt}/\hspace{-1.111pt}\psi }} $$ J / ψ ($$\rightarrow $$ $$\mu ^+\mu ^-$$ μ + μ - )$$\pi ^+\pi ^-$$ π + π - final state in the forward region using proton-proton collision data collected by the LHCb experiment at the center-of-mass-energy of$$13\text {TeV} $$ 13 TeV in 2016, corresponding to an integrated luminosity of$$1.64\,\text {\,fb} ^{-1} $$ 1.64 \,fb - 1 . The fragmentation function, presented as the ratio of the quarkonium-tag transverse momentum to the full jet transverse momentum ($$p_{\textrm{T}} (\text {tag})/p_{\textrm{T}} (\text {jet})$$ p T ( tag ) / p T ( jet ) ), is measured differentially in$$p_{\textrm{T}} (\text {jet})$$ p T ( jet ) and$$p_{\textrm{T}} (\text {tag})$$ p T ( tag ) bins. The distributions are separated into promptly produced quarkonia from proton-proton collisions and quarkonia produced from displacedb-hadron decays. While the displaced quarkonia fragmentation functions are in general well described by parton-shower predictions, the prompt quarkonium distributions differ significantly from fixed-order non-relativistic QCD (NRQCD) predictions followed by a QCD parton shower. 
    more » « less
  5. Abstract Given$$g \in \mathbb N \cup \{0, \infty \}$$ g N { 0 , } , let$$\Sigma _g$$ Σ g denote the closed surface of genusgwith a Cantor set removed, if$$g<\infty $$ g < ; or the blooming Cantor tree, when$$g= \infty $$ g = . We construct a family$$\mathfrak B(H)$$ B ( H ) of subgroups of$${{\,\textrm{Map}\,}}(\Sigma _g)$$ Map ( Σ g ) whose elements preserve ablock decompositionof$$\Sigma _g$$ Σ g , andeventually like actlike an element ofH, whereHis a prescribed subgroup of the mapping class group of the block. The group$$\mathfrak B(H)$$ B ( H ) surjects onto an appropriate symmetric Thompson group of Farley–Hughes; in particular, it answers positively. Our main result asserts that$$\mathfrak B(H)$$ B ( H ) is of type$$F_n$$ F n if and only ifHis. As a consequence, for every$$g\in \mathbb N \cup \{0, \infty \}$$ g N { 0 , } and every$$n\ge 1$$ n 1 , we construct a subgroup$$G <{{\,\textrm{Map}\,}}(\Sigma _g)$$ G < Map ( Σ g ) that is of type$$F_n$$ F n but not of type$$F_{n+1}$$ F n + 1 , and which contains the mapping class group of every compact surface of genus$$\le g$$ g and with non-empty boundary. 
    more » « less