Character Values of Stanley Sequences

Stanley and Odlyzko proposed a method for greedily constructing sets with no 3-term arithmetic progressions. It is conjectured that there is a dichotomy between such sequences: those that have a periodic structure as the sequence satisfies certain recurrence relations while others appear to be chaotic. One large class of sequences that have these periodic behaviors are known as independent sequences that have two parameters, a character and a growth factor. It was conjectured by Rolnick that all but a finite set of integers can be achieved as characters of a independent sequences. Previously the only large class of integers known to be characters where those with base 3 representations consisting solely of the digits 0 and 2. This paper dramatically improves this result by demonstrating that all even integers not congruent to 244 mod 486 can be achieved as characters, therefore demonstrating that the set of all characters has a positive lower density.Comment: Java Code for Verification Adde

On the classification of Stanley sequences

An integer sequence is said to be 3-free if no three elements form an arithmetic progression. Following the greedy algorithm, the Stanley sequence $S(a_0,a_1,\ldots,a_k)$ is defined to be the 3-free sequence $\{a_n\}$ having initial terms $a_0,a_1,\ldots,a_k$ and with each subsequent term $a_n>a_{n-1}$ chosen minimally such that the 3-free condition is not violated. Odlyzko and Stanley conjectured that Stanley sequences divide into two classes based on asymptotic growth patterns, with one class of highly structured sequences satisfying $a_n\approx \Theta(n^{\log_2 3})$ and another class of seemingly chaotic sequences obeying $a_n=\Theta(n^2/\log n)$. We propose a rigorous definition of regularity in Stanley sequences based on local structure rather than asymptotic behavior and show that our definition implies the corresponding asymptotic property proposed by Odlyzko and Stanley. We then construct many classes of regular Stanley sequences, which include as special cases all such sequences previously identified. We show how two regular sequences may be combined into another regular sequence, and how parts of a Stanley sequence may be translated while preserving regularity. Finally, we demonstrate that certain Stanley sequences possess proper subsets that are also Stanley sequences, a situation that appears previously to have been assumed impossible.Comment: 25 page

A Demazure crystal construction for Schubert polynomials

Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur functions and Demazure characters for the general linear group. We establish this connection by imposing a Demazure crystal structure on key tableaux, recently introduced by the first author in connection with Demazure characters and Schubert polynomials, and linking this to the type A crystal structure on reduced word factorizations, recently introduced by Morse and the second author in connection with Stanley symmetric functions.Comment: 18 pages, 16 figures; version 2: references added and update

Playing with Identity. Authors, Narrators, Avatars, and Players in The Stanley Parable and The Beginner’s Guide

This article offers a comparative analysis of Davey Wreden’s The Stanley Parable (Wreden 2011 / Galactic Cafe 2013) and The Beginner’s Guide (Everything Unlimited Ltd. 2015) in order to explore the interrelation of authors, narrators, avatars, and players as four salient functions in the play with identity that videogames afford. Building on theories of collective and collaborative authorship, of narratives and narrators across media, and of the avatar-player relationship, the article reconstructs the similarities and differences between the way in which The Stanley Parable and The Beginner’s Guide position their players in relation to the two games’ avatars, narrators, and (main) author, while also underscoring how both The Stanley Parable and The Beginner’s Guide use metareferential strategies to undermine any overly rigid conceptualization of these functions and their interrelation

Comparison of articulate brachiopod nuclear and mitochondrial gene trees leads to a clade-based redefinition of protostomes (Protostomozoa) and deuterostomes (Deuterostomozoa)

Nuclear and mtDNA sequences from selected short-looped terebratuloid (terebratulacean) articulate brachiopods yield congruent and genetically independent phylogenetic reconstructions by parsimony, neighbor-joining and maximum likelihood methods, suggesting that both sources of data are reliable guides to brachiopod species phylogeny. The present-day genealogical relationships and geographical distributions of the tested terebratuloid brachiopods are consistent with a tethyan dispersal and subsequent radiation. Concordance of nuclear and mitochondrial gene phylogenies reinforces previous indications that articulate brachiopods, inarticulate brachiopods, phoronids and ectoprocts cluster with other organisms generally regarded as protostomes. Since ontogeny and morphology in brachiopods, ectoprocts and phoronids depart in important respects from those features supposedly diagnostic of protostomes, this demonstrates that the operational definition of protostomy by the usual ontological characters must be misleading or unreliable. New, molecular, operational definitions are proposed to replace the traditional criteria for the recognition of protostomes and deuterostomes, and the clade-based terms 'Protostomozoa' and 'Deuterostomozoa' are proposed to replace the existing terms 'Protostomia' and 'Deuterostomia'