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Fix an integer partition lambda that has no more than n parts. Let beta be a weakly increasing n-tuple with entries from {1,..,n}. The flagged Schur function indexed by lambda and beta is a polynomial generating function in x_1, .., x_n for certain semistandard tableaux of shape lambda. Let pi be an n-permutation. The type A Demazure character (key polynomial, Demazure polynomial) indexed by lambda and pi is another such polynomial generating function. Reiner and Shimozono and then Postnikov and Stanley studied coincidences between these two families of polynomials. Here their results are sharpened by the specification of unique representatives for the equivalence classes of indexes for both families of polynomials, extended by the consideration of more general beta, and deepened by proving that the polynomial coincidences also hold at the level of the underlying tableau sets. Let R be the set of lengths of columns in the shape of lambda that are less than n. Ordered set partitions of {1,..,n} with block sizes determined by R, called R-permutations, are used to describe the minimal length representatives for the parabolic quotient of the nth symmetric group specified by the set {1,..,n-1}\R. The notion of 312-avoidance is generalized from n-permutations to these set partitions. The R-parabolic Catalan number is defined to be the number of these. Every flagged Schur function arises as a Demazure polynomial. Those Demazure polynomials are precisely indexed by the R-312-avoiding R-permutations. Hence the number of flagged Schur functions that are distinct as polynomials is shown to be the R-parabolic Catalan number. The projecting and lifting processes that relate the notions of 312-avoidance and of R-312-avoidance are described with maps developed for other purposes., Comment: 27 pages, 2 figures. Identical to v.2, except for the insertion of the publication data for the DMTCS journal (dates and volume/issue/number). This is two-thirds of our preprint "Parabolic Catalan numbers count flagged Schur functions; Convexity of tableau sets for Demazure characters", arXiv:1612.06323v2

This is the first of three papers that develop structures which are counted by a "parabolic" generalization of Catalan numbers. Fix a subset R of {1,..,n-1}. Consider the ordered partitions of {1,..,n} whose block sizes are determined by R. These are the "inverses" of (parabolic) multipermutations whose multiplicities are determined by R. The standard forms of the ordered partitions are refered to as "R-permutations". The notion of 312-avoidance is extended from permutations to R-permutations. Let lambda be a partition of N such that the set of column lengths in its shape is R or R union {n}. Fix an R-permutation pi. The type A Demazure character (key polynomial) in x_1, .., x_n that is indexed by lambda and pi can be described as the sum of the weight monomials for some of the semistandard Young tableau of shape lambda that are used to describe the Schur function indexed by lambda. Descriptions of these "Demazure" tableaux developed by the authors in earlier papers are used to prove that the set of these tableaux is convex in Z^N if and only if pi is R-312-avoiding if and only if the tableau set is the entire principal ideal generated by the key of pi. These papers were inspired by results of Reiner and Shimozono and by Postnikov and Stanley concerning coincidences between Demazure characters and flagged Schur functions. This convexity result is used in the next paper to deepen those results from the level of polynomials to the level of tableau sets. The R-parabolic Catalan number is defined to be the number of R-312-avoiding permutations. These special R-permutations are reformulated as "R-rightmost clump deleting" chains of subsets of {1,..,n} and as "gapless R-tuples"; the latter n-tuples arise in multiple contexts in these papers., Comment: 20 pp with 2 figs. Identical to v.3, except for the insertion of the publication data for the DMTCS journal (dates and volume/issue/number). This is one third of our "Parabolic Catalan numbers ..", arXiv:1612.06323v2

Although d-complete posets arose along the interface between algebraic combinatorics and Lie theory, they are defined using only requirements on their local structure. These posets are a mutual generalization of rooted trees, shapes, and shifted shapes. They possess Stanley's hook product property for their P-partition generating functions and Schutzenberger's well defined jeu de taquin rectification property. The original definition of d-complete poset was lengthy, but more succinct definitions were later developed. Here several definitions are shown to be equivalent. The basic properties of d-complete posets are summarized. Background and a partial bibliography for these posets is given., Comment: 26 pages, 5 figures. On the question of how to define the notion of "d-complete" for infinite posets: A sentence on p. 6 is reworded, the last sentence on p. 16 is deleted, and an Added Note is inserted. References [KiYo] and [NrOk] are new. This is the final uncopyrighted version; the 'Order' journal version will also include several small exposition clarifications from the referee

Let $\lambda$ be a partition with no more than $n$ parts. Let $\beta$ be a weakly increasing $n$-tuple with entries from $\{ 1, ... , n \}$. The flagged Schur function in the variables $x_1, ... , x_n$ that is indexed by $\lambda$ and $\beta$ has been defined to be the sum of the content weight monomials for the semistandard Young tableaux of shape $\lambda$ whose values are row-wise bounded by the entries of $\beta$. Gessel and Viennot gave a determinant expression for the flagged Schur function indexed by $\lambda$ and $\beta$; this could be done since the pair $(\lambda, \beta)$ satisfied their "nonpermutable" condition for the sequence of terminals of an $n$-tuple of lattice paths that they used to model the tableaux. We generalize flagged Schur functions by dropping the requirement that $\beta$ be weakly increasing. Then for each $\lambda$ we give a condition on the entries of $\beta$ for the pair $(\lambda, \beta)$ to be nonpermutable that is both necessary and sufficient. When the parts of $\lambda$ are not distinct there will be multiple row bound $n$-tuples $\beta$ that will produce the same set of tableaux. We accordingly group the bounding $\beta$ into equivalence classes and identify the most efficient $\beta$ in each class for the determinant computation. We recently showed that many other sets of objects that are indexed by $n$ and $\lambda$ are enumerated by the number of these efficient $n$-tuples. We called these counts "parabolic Catalan numbers". It is noted that the $GL(n)$ Demazure characters (key polynomials) indexed by 312-avoiding permutations can also be expressed with these determinants., Comment: 22 pages, 5 figures, 4 tables. Identical to v.5, except for the insertion of a reference and the DMTCS journal's publication meta data

Shuffles are n-multipermutations with suit multiplicities given by a subset R of {1,..,n-1}. Their inverses are ordered partitions of {1,..,n} whose block sizes derive from R. These "R-permutations" depict the min length coset reps for the quotient of S_n by the parabolic subgroup W_J, with J the complement of R. We refer to those that blockwise avoid the pattern 312 as "312-avoiding R-permutations" and define the "parabolic R-Catalan number" to be the number of them. Let lambda be a partition of N with at most n parts whose set of shape column lengths less than n is R. We show that the number of flagged Schur functions formed on the shape of lambda is this parabolic R-Catalan number, and list over a dozen other phenomena that are enumerated by it. Let pi be an R-permutation. We view the Demazure character (key polynomial) indexed by (lambda,pi) as the sum of the content weight monomials for our "pi-Demazure" semistandard tableaux of shape lambda with entries from {1,..,n}. We show that the set of these tableaux is convex in Z^N if and only if pi is a 312-avoiding R-permutation. A flagged Schur function is the sum of the content weight monomials for the semistandard tableaux of shape lambda whose entries are row-wise bounded by a given weakly increasing n-tuple. We consider general row bound sums for which the bounds may be any n-tuple. Reiner and Shimozono and then Postnikov and Stanley obtained results concerning coincidences between flagged Schur functions and Demazure characters: when lambda is strict, the flagged Schur functions exactly coincide with the 312-avoiding Demazure characters. For general lambda we introduce precise indexing sets of n-tuple bounds for the row bound sums. This and our convexity results are used to sharpen their coincidence results, to extend them to general row bound sums, and to show they hold at the deeper level of coinciding underlying tableau sets., Comment: 49 pages with 2 figures, plus a 2 page table of symbols for 51 pages total. Inserted figures for two proofs, appended Added Note, appended summary paragraph to introduction, merged Corollary 11.3 into Theorem 11.1, and polished here and there

Globular clusters (GCs) can be considered discrete, long-lived, dynamical tracers that retain crucial information about the assembly history of their parent galaxy. In this paper, we present a new catalogue of GC velocities and colours for the lenticular galaxy NGC 1023, we study their kinematics and spatial distribution, in comparison with the underlying stellar kinematics and surface brightness profile, and we test a new method for studying GC properties. Specifically, we decompose the galaxy light into its spheroid (assumed to represent the bulge + halo components) and disk components and use it to assign to each GC a probability of belonging to one of the two components. Then we model the galaxy kinematics, assuming a disk and spheroidal component, using planetary nebulae (PNe) and integrated stellar light. We use this kinematic model and the probability previously obtained from the photometry to recalculate for each GC its likelihood of being associated with the disk, the spheroid, or neither. We find that the reddest GCs are likely to be associated with the disk, as found for faint fuzzies in this same galaxy, suggesting that the disk of this S0 galaxy originated at z ~ 2. The majority of blue GCs are found likely to be associated with the spheroidal (hot) component. The method also allows us to identify objects that are unlikely to be in equilibrium with the system. In NGC1023 some of the rejected GCs form a substructure in phase space that is connected with NGC 1023 companion galaxy., Comment: 12 pages, 10 figures, accepted MNRAS

We report the results of a multi-waveband analysis of the masses and luminosities of $\sim$600 galaxy groups and clusters identified in the maxBCG catalogue. These data are intended to form the basis of future work on the formation of the "$m_{12}$ gap" in galaxy groups and clusters. We use SDSS spectroscopy and $g$, $r$ and $i$ band photometry to estimate galaxy group/cluster virial radii, masses and total luminosities. In order to establish the robustness of our results, we compare them with literature studies that utilize a variety of mass determinations techniques (dynamical, X-ray, weak lensing) and total luminosities estimated in the $B$, $r$, $i$, and $K$ wavebands. We also compare our results to predictions derived from the Millennium Simulation. We find that, once selection effects are properly accounted for, excellent agreement exists between our results and the literature with the exception of a single observational study. We also find that the Millennium Simulation does an excellent job of predicting the effects of our selection criteria. Our results show that, over the mass range $\sim10^{13}-10^{15}$ M$_{\odot}$, variations in the slope of the mass-luminosity scaling relation with mass detected in this and many other literature studies is in part the result of selection effects. We show that this can have serious ramifications on attempts to determine how the mass-to-light ratio of galaxy groups and cluster varies with mass., Comment: 10 pages, 5 figures, Accepted for publication in MNRAS

We report the 6dFGS Fundamental Plane (6dFGSv) catalogue that is used to estimate distances and peculiar velocities for nearly 9,000 early-type galaxies in the local (z$<$0.055) universe. Velocity dispersions are derived by cross-correlation from 6dF V-band spectra with typical S/N of 12.9 \AA$^{-1}$ for a sample of 11,315 galaxies; the median velocity dispersion is 163 kms$^{-1}$ and the median measurement error is 12.9%. The photometric Fundamental Plane (FP) parameters (effective radii and surface brightnesses) are determined from the $JHK$ 2MASS images for 11,102 galaxies. Comparison of the independent $J$- and $K$-band measurements implies that the average uncertainty in $X_{FP}$, the combined photometric parameter that enters the FP, is 0.013 dex (3%) for each band. Visual classification of morphologies was used to select a sample of nearly 9,000 early-type galaxies that form 6dFGSv. This catalogue has been used to study the effects of stellar populations on galaxy scaling relations, to investigate the variation of the FP with environment and galaxy morphology, to explore trends in stellar populations through, along and across the FP, and to map and analyse the local peculiar velocity field., Comment: 21 pages, 25 figures, 9 tables, accepted for publication in MNRAS, the full versions of Tables 2, 4, 8 and 9 will be available in the MNRAS publication as online Supporting Information

We report the results of Gemini/GMOS long-slit spectroscopic observations along the major and minor axes of the central galaxies in two fossil groups, SDSS J073422.21+265133.9 and SDSS J075828.11+374711.8 (the NGC 2484 group). Spatially resolved kinematics and stellar population parameters (ages, metallicities and $\alpha$-element abundance ratios) derived using ~20 Lick indices are presented. Despite remarkable similarities in their morphologies, photometric properties (luminosity and colour) and kinematics, the two galaxies exhibit significantly different stellar population parameters. SDSS J073422.21+265133.9 exhibits a strong metallicity gradient (Delta[Z/H]/Delta R ~ -0.4) all the way into the centre of the galaxy. It also exhibits an age profile that suggest a relatively recent, centrally concentrated burst of star formation superimposed on an older, more spatially extended population. NGC 2484, a well known X-ray AGN, exhibits a flat core-like structure in its metallicity gradient, but no detectable age gradient. The alpha-element abundance ratio ([E/Fe]) profiles of the two galaxies are also significantly different. SDSS J073422.21+265133.9 exhibits a slightly positive gradient (Delta [E/H]/Delta R ~ 0.1), perhaps again suggesting a more recent central burst of star formation, while NGC 2484 shows a negative gradient (Delta [E/H]/Delta R ~-0.1), indicating that star formation may have happened "inside out". Our analysis of these two galaxies of similar mass, morphology and kinematics therefore suggests two different mechanisms to have been in action during their formation. Consequently, we conclude that the central galaxies of fossil groups can not be considered a homogeneous group with regard to their formation processes or star formation histories., Comment: 10 pages, 4 figures. Accepted for Publication in MNRAS

The Schur function indexed by a partition lambda with at most n parts is the sum of the weight monomials for the Young tableaux of shape lambda. Let pi be an n-permutation. We give two descriptions of the tableaux that contribute their monomials to the key polynomial indexed by pi and lambda. (These polynomials are the characters of the Demazure modules for GL(n).) The "atom" indexed by pi is the sum of weight monomials of the tableaux whose right keys are the "key" tableau for pi. Schur functions and key polynomials can be decomposed into sums of atoms. We also describe the tableaux that contribute to an atom, the tableaux that have a left key equal to a given key, and the tableaux that have a left key bounded below by a given key., Comment: 19 pages w/ 3 figures (1 new), 1 page with post-publication minor improvements. New title. This is the final pre-typesetting version of the European Journal of Combinatorics paper 43 (2015), 172-184. This v3 has many revisions and clarifications that were inspired by the referees' reports. These include Proposition 1.1, plus the added hypothesis for Theorem 3.2 and for Corollaries 10.2 and 10.6

This paper presents further results from our spectroscopic study of the globular cluster (GC) system of the group elliptical NGC 3923. From observations made with the GMOS instrument on the Gemini South telescope, an additional 50 GC and Ultra Compact Dwarf (UCD) candidates have been spectroscopically confirmed as members of the NGC 3923 system. When the recessional velocities of these GCs are combined with the 29 GC velocities reported previously, a total sample of 79 GC/UCD velocities is produced. This sample extends to over 6 arcmin (>6 Re \sim30 kpc) from the centre of NGC 3923, and is used to study the dynamics of the GC system and the dark matter content of NGC 3923. It is found that the GC system of NGC 3923 displays no appreciable rotation, and that the projected velocity dispersion is constant with radius within the uncertainties. The velocity dispersion profiles of the integrated light and GC system of NGC 3923 are indistinguishable over the region in which they overlap. We find some evidence that the diffuse light and GCs of NGC 3923 have radially biased orbits within \sim130". The application of axisymmetric orbit-based models to the GC and integrated light velocity dispersion profiles demonstrates that a significant increase in the mass-to-light ratio (from M/Lv = 8 to 26) at large galactocentric radii is required to explain these observations. We therefore confirm the presence of a dark matter halo in NGC 3923. We find that dark matter comprises 17.5% of the mass within 1 Re, 41.2% within 2 Re, and 75.6% within the radius of our last kinematic tracer at 6.9 Re. The total dynamical mass within this radius is found to be 1.5 x 10^12 solar masses. In common with other studies of large ellipticals, we find that our derived dynamical mass profile is consistently higher than that derived by X-ray observations, by a factor of around 2., Comment: 17 pages, 10 figures, accepted for publication in MNRAS

We report the discovery of 12 new fossil groups of galaxies, systems dominated by a single giant elliptical galaxy and cluster-scale gravitational potential, but lacking the population of bright galaxies typically seen in galaxy clusters. These fossil groups (FGs), selected from the maxBCG optical cluster catalog, were detected in snapshot observations with the Chandra X-ray Observatory. We detail the highly successful selection method, with an 80% success rate in identifying 12 FGs from our target sample of 15 candidates. For 11 of the systems, we determine the X-ray luminosity, temperature, and hydrostatic mass, which do not deviate significantly from expectations for normal systems, spanning a range typical of rich groups and poor clusters of galaxies. A small number of detected FGs are morphologically irregular, possibly due to past mergers, interaction of the intra-group medium (IGM) with a central AGN, or superposition of multiple massive halos. Two-thirds of the X-ray-detected FGs exhibit X-ray emission associated with the central BCG, although we are unable to distinguish between AGN and extended thermal galaxy emission using the current data. This sample, a large increase in the number of known FGs, will be invaluable for future planned observations to determine FG temperature, gas density, metal abundance, and mass distributions, and to compare to normal (non-fossil) systems. Finally, the presence of a population of galaxy-poor systems may bias mass function determinations that measure richness from galaxy counts. When used to constrain power spectrum normalization and {\Omega}_m, these biased mass functions may in turn bias these results., Comment: 15 pages, 8 figures; changes reflect the version accepted to ApJ

Defined as X-ray bright galaxy groups with large differences between the luminosities of their brightest and second brightest galaxies, "fossil groups" are believed to be some of the oldest galaxy systems in the universe. They have therefore been the subject of much recent research. In this work we present a study of 10 fossil group candidates with an average of 33 spectroscopically confirmed members per group, making this the deepest study of its type to-date. We also use this data to perform an analysis of the luminosity function of our sample of fossil groups. We confirm the high masses previously reported for many of fossil systems, finding values more similar to those of clusters than of groups. We also confirm the high dynamical mass-to-light ratios reported in many previous studies. While our results are consistent with previous studies in many ways, our interpretation is not. This is because we show that, while the luminosities of the BCGs in these systems are consistent with their high dynamical masses, their richnesses (total number of galaxies above some canonical value) are extremely low. This leads us to suggest a new interpretation of fossil systems in which the large differences between the luminosities of their brightest and second brightest galaxies are simply the result the high BCG luminosities and low richnesses, while the high masses and low richnesses also explain the high mass-to-light ratios. Our results therefore suggest that fossil systems can be characterised as cluster-like in their masses and BCG luminosities, but possessing the richnesses and optical luminosities of relatively poor groups. These findings are not predicted by any of the current models for the formation of fossil groups. Therefore, if this picture is confirmed, current ideas about the formation and evolution of fossil systems will need to be reformulated., Comment: 22 pages, 8 figures. Accepted for publication in MNRAS

We present a comprehensive analysis of the spatial, kinematic, and chemical properties of stars and globular clusters (GCs) in the `ordinary' elliptical galaxy NGC 4494 using data from the Keck and Subaru telescopes. We derive galaxy surface brightness and colour profiles out to large galactocentric radii. We compare the latter to metallicities derived using the near-infrared Calcium Triplet. We obtain stellar kinematics out to ~3.5 effective radii. The latter appear flattened or elongated beyond ~1.8 effective radii in contrast to the relatively round photometric isophotes. In fact, NGC 4494 may be a flattened galaxy, possibly even an S0, seen at an inclination of ~45 degrees. We publish a catalogue of 431 GC candidates brighter than i=24 based on the photometry, of which 109 are confirmed spectroscopically and 54 have measured spectroscopic metallicities. We also report the discovery of 3 spectroscopically confirmed ultra-compact dwarfs around NGC 4494 with measured metallicities of -0.4<[Fe/H]<-0.3. Based on their properties, we conclude that they are simply bright GCs. The metal-poor globular clusters are found to be rotating with similar amplitude as the galaxy stars, while the metal-rich globular clusters show marginal rotation. We supplement our analysis with available literature data and results. Using model predictions of galaxy formation, and a suite of merger simulations, we find that many of the observational properties of NGC 4494 may be explained by formation in a relatively recent gas-rich major merger. Complete studies of individual galaxies incorporating a range of observational avenues and methods such as the one presented here will be an invaluable tool for constraining the fine details of galaxy formation models, especially at large galactocentric radii., Comment: 23 pages, 18 figures, accepted for publication in MNRAS