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We report the first direct measurement of the nuclear electron capture (EC) decay $Q$-value of $^{7}$Be $\rightarrow$ $^{7}$Li via high-precision Penning trap mass spectrometry (PTMS). This was performed using the LEBIT Penning trap located at the National Superconducting Cyclotron Laboratory/Facility for Rare Isotope Beams (NSCL/FRIB) using the newly commissioned Batch-Mode Ion-Source (BMIS) to deliver the unstable $^{7}$Be$^{+}$ samples. With a measured value of $Q_{EC}$ = 861.963(23) keV this result is also three times more precise than any previous determination of this quantity. This improved precision, and accuracy of the $^7$Be EC decay $Q$-value is critical for ongoing experiments that measure the recoiling nucleus in this system as a signature to search for beyond Standard Model (BSM) neutrino physics using $^7$Be-doped superconducting sensors.

Light curves are the primary observable of type-I x-ray bursts. Computational x-ray burst models must match simulations to observed light curves. Most of the error in simulated curves comes from uncertainties in $rp$ process reaction rates, which can be reduced via precision mass measurements of neutron-deficient isotopes in the $rp$ process path. We perform a precise Penning trap mass measurement of $^{27}$P utilizing the ToF-ICR technique. We use this measurement to calculate $rp$ process reaction rates and input these rates into an x-ray burst model to reduce simulated light curve uncertainty. We also use the mass measurement of $^{27}$P to validate the Isobaric Multiplet Mass Equation (IMME) for the A=27 T=$\frac{3}{2}$ isospin quartet which $^{27}$P belongs to. The mass excess of $^{27}$P was measured to be -670.7(6) keV, a fourteen-fold precision increase over the mass reported in the 2020 Atomic Mass Evaluation (AME2020). X-ray burst light curves were produced with the MESA (Modules for Experiments in Stellar Astrophysics) code using the new mass and associated reaction rates. Changes in the mass of $^{27}$P seem to have minimal effect on light curves, even in burster systems tailored to maximize impact. The mass of $^{27}$P does not play a significant role in x-ray burst light curves. It is important to understand that more advanced models do not just provide more precise results, but often qualitatively different ones. This result brings us a step closer to extracting stellar parameters from individual x-ray burst observations. The IMME has been validated for the $A=27, T=3/2$ quartet. The normal quadratic form of the IMME using the latest data yields a reduced $\chi^2$ of 2.9. The cubic term required to generate an exact fit to the latest data matches theoretical attempts to predict this term., Comment: 8 pages, 6 figures

Background: Low energy $\beta$ and electron capture (EC) decays are important systems in neutrino mass determination experiments. An isotope with an ultra-low Q value $\beta$-decay to an excited state in the daughter with Qes < 1 keV could provide a promising alternative candidate for future experiments. $^{75}$Se EC and $^{75}$Ge $\beta$-decay represent such candidates, but a more precise determination of the mass of the common daughter, $^{75}$As, is required to evaluate whether their potential decay branches are energetically allowed and ultra-low. Purpose: Perform a precise atomic mass measurement of $^{75}$As and combine the result with the precisely known atomic masses of $^{75}$Se and $^{75}$Ge, along with nuclear energy level data for $^{75}$As to evaluate potential ultra-low Q value decay branches in the EC decay of $^{75}$Se and the $\beta$-decay of $^{75}$Ge. Method: The LEBIT Penning trap mass spectrometer at the Facility for Rare Isotope Beams was used to perform a high-precision measurement of the atomic mass of $^{75}$As via cyclotron frequency ratio measurements of $^{75}$As$^{+}$ to a $^{12}$C$_{6}^{+}$ reference ion. Results: The $^{75}$As mass excess was determined to be ME($^{75}$As)= -73 035.98(43) keV, from which the ground-state to ground-state Q values for $^{75}$Se EC and $^{75}$Ge $\beta$-decay were determined to be 866.50(44) keV and 1179.01(44) keV, respectively. These results were compared to energies of excited states in $^{75}$As at 865.4(5) keV and 1172.0(6) keV to determine Q values of 1.1(7) keV and 7.0(7) keV for the potential ultra-low EC and $\beta$-decay branches of $^{75}$Se and $^{75}$Ge, respectively. Conclusion: The $^{75}$Se EC decay to the 865.4 keV excited state in $^{75}$As is potentially ultra-low with Qes $\approx$ 1 keV. However, a more precise determination of the 865.4(5) keV level in $^{75}$As is required., Comment: 7 pages,3 figures

Mainly because of its long half-life and despite its scientific relevance, spectroscopic measurements of $^{176}$Lu forbidden $\beta$-decays are very limited and lack formulation of shape factors. A direct precise measurement of its $Q$ value is also presently unreported. In addition, the description of forbidden decays provides interesting challenges for nuclear theory. The comparison of precise experimental results with theoretical calculations for these decays can help to test underlying models and can aid the interpretation of data from other experiments.

We report a high precision mass measurement of $^{24}{\rm Si}$, performed with the LEBIT facility at the National Superconducting Cyclotron Laboratory. The atomic mass excess, $10\;753.8$(37) keV, is a factor of 5 more precise than previous results. This substantially reduces the uncertainty of the $^{23}{\rm Al}(p,\gamma)^{24}{\rm Si}$ reaction rate, which is a key part of the rapid proton capture ($rp$) process powering Type I X-ray bursts. The updated rate constrains the onset temperature of the $(\alpha,p)$ process at the $^{22}{\rm Mg}$ waiting-point to a precision of 9%., Comment: 6 pages, 3 figures

Pipe dreams and bumpless pipe dreams for vexillary permutations are each known to be in bijection with certain semistandard tableaux via maps due to Lenart and Weigandt, respectively. Recently, Gao and Huang have defined a bijection between the former two sets. In this note we show for vexillary permutations that the Gao-Huang bijection preserves the associated tableaux, giving a new proof of Lenart's result. Our methods extend to give a recording tableau for any bumpless pipe dream., Comment: 19 pages including 6 page appendix

Nuclear mass measurements of isotopes are key to improving our understanding of nuclear structure across the chart of nuclides, in particular for the determination of the appearance or disappearance of nuclear shell closures. We present high-precision mass measurements of neutron-rich Ca, Ti and V isotopes performed at the TITAN and LEBIT facilities. These measurements were made using the TITAN multiple-reflection time-of-flight mass spectrometer (MR-ToF-MS) and the LEBIT 9.4T Penning trap mass spectrometer. In total, 13 masses were measured, eight of which represent increases in precision over previous measurements. These measurements refine trends in the mass surface around $N = 32$ and $N = 34$, and support the disappearance of the $N = 32$ shell closure with increasing proton number. Additionally, our data does not support the presence of a shell closure at $N = 34$., Comment: 8 pages, 6 figures

The goal of this monograph is to study the indicator function for a set of permutations mapping one finite sequence of positive integers to another from a representation theoretic, combinatorial and probabilistic perspective. The degree of a function of permutations is the size of the largest pair of sequences required when expressing it as a linear combination of these indicators. This notion of degree, implicit in work of Diaconis, is critical for many applications of representation theory to extremal combinatorics, machine learning, probability and statistics. We use the term local to indicate bounded degree and initiate the study of low degree class functions, which encode probabilistic data for permutation statistics on each cycle type simultaneously. We begin with a self contained treatment of for functions of permutations, developing its theory using language familiar to enumerative and algebraic combinatorialists. This leads naturally to a novel basis for symmetric functions we call the path power sum symmetric functions. The most technically challenging part of our work is the path Murnaghan-Nakayama formula, which expands path power sums into Schur functions. By combining the the path Murnaghan-Nakayam formula with the classical theory of character polynomials, one obtains a structural characterization for moments for permutation statistics conditioning on cycle type. We then analyze asymptotic properties of these moments. In doing so, we introduce the novel family of regular permutation statistics, which include almost all reasonable weighted pattern counting statistics. We show a large family of regular statistics satisfy a law of large numbers on a given cycle type depending only on the proportion of fixed points and a have variances depending only on fixed points and two cycles., Comment: 131 pages, 14 figures

In the first partial result toward Steinberg's now-disproved three coloring conjecture, Abbott and Zhou used a counting argument to show that every planar graph without cycles of lengths 4 through 11 is 3-colorable. Implicit in their proof is a fact about plane graphs: in any plane graph of minimum degree 3, if no two triangles share an edge, then triangles make up strictly less than 2/3 of the faces. We show how this result, combined with Kostochka and Yancey's resolution of Ore's conjecture for k = 4, implies that every planar graph without cycles of lengths 4 through 8 is 3-colorable.

We performed a Penning trap mass measurement of $^{61}{\rm Zn}$ at the National Superconducting Cyclotron Laboratory and NuShellX calculations of the $^{61}{\rm Zn}$ and $^{62}{\rm Ga}$ structure using the GXPF1A Hamiltonian to obtain improved estimates of the $^{61}{\rm Zn}(p,\gamma)^{62}{\rm Ga}$ and $^{60}{\rm Cu}(p,\gamma)^{61}{\rm Zn}$ reaction rates. Surveying astrophysical conditions for type-I X-ray bursts with the code MESA, implementing our improved reaction rates, and taking into account updated nuclear masses for $^{61}{\rm V}$ and $^{61}{\rm Cr}$ from the recent literature, we refine the neutrino luminosity from the important mass number $A=61$ urca cooling source in accreted neutron star crusts. This improves our understanding of the thermal barrier between deep heating in the crust and the shallow depths where extra heat is needed to explain X-ray superbursts, as well as the expected signature of crust urca neutrino emission in light curves of cooling transients., Comment: Submitted

Protons and neutrons in the atomic nucleus move in shells analogous to the electronic shell structures of atoms. Nuclear shell structure varies across the nuclear landscape due to changes of the nuclear mean field with the number of neutrons $N$ and protons $Z$. These variations can be probed with mass differences. The $N=Z=40$ self-conjugate nucleus $^{80}$Zr is of particular interest as its proton and neutron shell structures are expected to be very similar, and its ground state is highly deformed. In this work, we provide evidence for the existence of a deformed double shell closure in $^{80}$Zr through high precision Penning trap mass measurements of $^{80-83}$Zr. Our new mass values show that $^{80}$Zr is significantly lighter, and thus more bound than previously determined. This can be attributed to the deformed shell closure at $N=Z=40$ and the large Wigner energy. Our statistical Bayesian model mixing analysis employing several global nuclear mass models demonstrates difficulties with reproducing the observed mass anomaly using current theory., Comment: 11 pages; 5 figures; submitted version, revised version accepted in Nature Physics pending minor revisions

In this note, we introduce a statistic on Motzkin paths that describes the rank generating function of Bruhat order for involutions. Our proof relies on a bijection introduced by Philippe Biane from permutations to certain labeled Motzkin paths and a recently introduced interpretation of this rank generating function in terms of visible inversions. By restricting our identity to fixed-point-free (FPF) involutions, we recover an identity due to Louis Billera, Lionel Levine and Karola M\'esz\'aros with a previous bijective proof by Matthew Watson. Our work sheds new light on the Ethiopian dinner game., Comment: 7 pages

We present a bijection between the set of standard Young tableaux of staircase minus rectangle shape, and the set of marked shifted standard Young tableaux of a certain shifted shape. Numerically, this result is due to DeWitt (2012). Combined with other known bijections this gives a bijective proof of the product formula for the number of standard Young tableaux of staircase minus rectangle shape. This resolves an open problem by Morales, Pak and Panova (2019), and allows for efficient random sampling. Other applications include a bijection for semistandard Young tableaux, and a bijective proof of Stembridge's symmetry of LR-coefficients of the staircase shape. We also extend these results to set-valued standard Young tableaux in the combinatorics of K-theory, leading to new proofs of results by Lewis and Marberg (2019) and Abney-McPeek, An and Ng (2020)., Comment: 19 pages, v3 corrects error in statement of what is now Theorem 2.11 plus other minor edits

We study the relationship between two notions of pattern avoidance for involutions in the symmetric group and their restriction to fixed-point-free involutions. The first is classical, while the second appears in the geometry of certain spherical varieties and generalizes the notion of pattern avoidance for perfect matchings studied by Jel\'inek. The first notion can always be expressed in terms of the second, and we give an effective algorithm to do so. We also give partial results characterizing the families of involutions where the converse holds. As a consequence, we prove two conjectures of McGovern characterizing (rational) smoothness of certain varieties. We also give new enumerative results, and conclude by proposing several lines of inquiry that extend our current work.

We report high-precision mass measurements of $^{50-55}$Sc isotopes performed at the LEBIT facility at NSCL and at the TITAN facility at TRIUMF. Our results provide a substantial reduction of their uncertainties and indicate significant deviations, up to 0.7 MeV, from the previously recommended mass values for $^{53-55}$Sc. The results of this work provide an important update to the description of emerging closed-shell phenomena at neutron numbers $N=32$ and $N=34$ above proton-magic $Z=20$. In particular, they finally enable a complete and precise characterization of the trends in ground state binding energies along the $N=32$ isotone, confirming that the empirical neutron shell gap energies peak at the doubly-magic $^{52}$Ca. Moreover, our data, combined with other recent measurements, does not support the existence of closed neutron shell in $^{55}$Sc at $N=34$. The results were compared to predictions from both \emph{ab initio} and phenomenological nuclear theories, which all had success describing $N=32$ neutron shell gap energies but were highly disparate in the description of the $N=34$ isotone.

Isobaric quintets provide the best test of the isobaric multiplet mass equation (IMME) and can uniquely identify higher order corrections suggestive of isospin symmetry breaking effects in the nuclear Hamiltonian. The Generalized IMME (GIMME) is a novel microscopic interaction theory that predicts an extension to the quadratic form of the IMME. Only the $A=20, 32$ $T=2$ quintets have the exotic $T_z = -2$ member ground state mass determined to high-precision by Penning trap mass spectrometry. In this work, we establish $A=36$ as the third high-precision $T=2$ isobaric quintet with the $T_z = -2$ member ground state mass measured by Penning trap mass spectrometry and provide the first test of the predictive power of the GIMME. A radioactive beam of neutron-deficient $^{36}$Ca was produced by projectile fragmentation at the National Superconducting Cyclotron Laboratory. The beam was thermalized and the mass of $^{36}$Ca$^+$ and $^{36}$Ca$^{2+}$ measured by the Time of Flight - Ion Cyclotron Resonance method in the LEBIT 9.4 T Penning trap. We measure the mass excess of $^{36}$Ca to be ME$ = -6483.6(56)$ keV, an improvement in precision by a factor of 6 over the literature value. The new datum is considered together with evaluated nuclear data on the $A=36$, $T=2$ quintet. We find agreement with the quadratic form of the IMME given by isospin symmetry, but only coarse qualitative agreement with predictions of the GIMME. A total of three isobaric quintets have their most exotic members measured by Penning trap mass spectrometry. The GIMME predictions in the $T = 2$ quintet appear to break down for $A = 32$ and greater.

The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by Knutson and Miller (2005) via antidiagonal Gr\"obner degeneration of matrix Schubert varieties. We consider instead diagonal Gr\"obner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics for the class of "vexillary'' matrix Schubert varieties. We initiate a study of general diagonal degenerations, relating them to a neglected formula of Lascoux (2002) in terms of the $6$-vertex ice model (recently rediscovered by Lam, Lee, and Shimozono (2018) in the guise of "bumpless pipe dreams'')., Comment: 22 pages

The Isobaric Multiplet Mass Equation (IMME) has been successful at predicting the masses of isobaric analogue states in the same multiplet, while its coefficients are known to follow trends as functions of mass number. The Atomic Mass Evaluation 2016 [Chin. Phys. C 41, 030003 (2017)] $^{44}$V mass value results in an negative $c$ coefficient for the IMME quadratic term. The $b$ and $c$ coefficients can provide constraints for construction of the isospin-nonconserving (INC) Hamiltonians for the $pf$ shell. The excitation energy of the $0^+, T=2$ level in $^{44}$V is currently unknown and can be used to constrain the $^{44}$Cr mass. The aim of the experiment was to perform high-precision mass measurements to resolve the difference between $^{44}$V isomeric and ground states, to test the IMME, and to provide ingredients for identifying the $0^+$, $T=2$ state in $^{44}$V. High-precision Penning trap mass spectrometry was performed at LEBIT, to measure the cyclotron frequency ratios of [$^{44g,m}$VO]$^+$ versus [$^{32}$SCO]$^+$, a reference mass, to extract both the isomeric and ground state masses of $^{44}$V. The mass excess of the ground and isomeric states in $^{44}$V were measured to be $-23\ 804.9(80)$ keV/$\text{c}^2$ and $-23\ 537.0(55)$ keV/$\text{c}^2$. This yielded a new proton separation energy of $S_p$ = 1\ 773(10) keV. The new mass values of $^{44}$V have been used to deduce the IMME $b$ and $c$ coefficients of the lowest $2^+$ and $6^+$ triplets in $A=44$. The $2^+$ $c$ coefficient is verified with the IMME trend and agrees with the shell-model predictions using charge-dependent Hamiltonians. The mirror energy differences were determined between $^{44}$V and $^{44}$Sc, in line with isospin-symmetry. The new value of the proton separation energy determined will be important for the determination of the $0^+$, $T=2$ state in $^{44}$V and for prediction of the mass of $^{44}$Cr., Comment: 9 pages, 5 figures