(s,t)-cores: a weighted version of Armstrong's conjecture

The study of core partitions has been very active in recent years, with the study of $(s,t)$-cores - partitions which are both $s$- and $t$-cores - playing a prominent role. A conjecture of Armstrong, proved recently by Johnson, says that the average size of an $(s,t)$-core, when $s$ and $t$ are coprime positive integers, is $\frac1{24}(s-1)(t-1)(s+t-1)$. Armstrong also conjectured that the same formula gives the average size of a self-conjugate $(s,t)$-core; this was proved by Chen, Huang and Wang. In the present paper, we develop the ideas from the author's paper [J. Combin. Theory Ser. A 118 (2011) 1525-1539] studying actions of affine symmetric groups on the set of $s$-cores in order to give variants of Armstrong's conjectures in which each $(s,t)$-core is weighted by the reciprocal of the order of its stabiliser under a certain group action. Informally, this weighted average gives the expected size of the $t$-core of a random $s$-core

2-chains: An interesting family of posets

We introduce a new family of finite posets which we call 2-chains. These first arose in the study of 0-Hecke algebras, but they admit a variety of different characterisations. We give these characterisations, prove that they are equivalent and derive some numerical results concerning 2-chains

Dyck tilings and the homogeneous Garnir relations for graded Specht modules

Suppose $\lambda$ and $\mu$ are integer partitions with $\lambda\supseteq\mu$. Kenyon and Wilson have introduced the notion of a cover-inclusive Dyck tiling of the skew Young diagram $\lambda\setminus\mu$, which has applications in the study of double-dimer models. We examine these tilings in more detail, giving various equivalent conditions and then proving a recurrence which we use to show that the entries of the transition matrix between two bases for a certain permutation module for the symmetric group are given by counting cover-inclusive Dyck tilings. We go on to consider the inverse of this matrix, showing that its entries are determined by what we call cover-expansive Dyck tilings. The fact that these two matrices are mutual inverses allows us to recover the main result of Kenyon and Wilson. We then discuss the connections with recent results of Kim et al, who give, a simple expression for the sum, over all $\mu$, of the number of cover-inclusive Dyck tilings of $\lambda\setminus\mu$. Our results provide a new proof of this result. Finally, we show how to use our results to obtain simpler expressions for the homogeneous Garnir relations for the universal Specht modules introduced by Kleshchev, Mathas and Ram for the cyclotomic quiver Hecke algebras

Crystals, regularisation and the Mullineux map

The Mullineux map is a combinatorial function on partitions which describes the effect of tensoring a simple module for the symmetric group in characteristic $p$ with the one-dimensional sign representation. It can also be interpreted as a signed isomorphism between crystal graphs for $\widehat{\mathfrak{sl}}_p$. We give a new combinatorial description of the Mullineux map by expressing this crystal isomorphism as a composition of isomorphisms between different crystals. These isomorphisms are defined in terms of new generalised regularisation maps introduced by Millan Berdasco. We then given two applications of our new realisation of the Mullineux map, by providing purely combinatorial proofs of a conjecture of Lyle relating the Mullineux map with regularisation, and a theorem of Paget describing the Mullineux map in RoCK blocks of symmetric groups

Evaluating three frameworks for the value of information: adaptation to task characteristics and probabilistic structure

We identify, and provide an integration of, three frameworks for measuring the informativeness of cues in a multiple-cue judgment task. Cues can be ranked by information value according to expected information gain (Bayesian framework), cue-outcome correlation (Correlational framework), or ecological validity (Ecological framework). In three experiments, all frameworks significantly predicted information acquisition, with the Correlational (then the Bayesian) framework being most successful. Additionally, participants adapted successfully to task characteristics (cue cost, time pressure, and information limitations) – altering the gross amount of information acquired, but not responding to more subtle features of the cues’ information value that would have been beneficial. Rational analyses of our task environments indicate that participants' behavior can be considered successful from a boundedly rational standpoint

Defect 2 spin blocks of symmetric groups and canonical basis coefficients

This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect 2 , analogous to Richards’s formula for defect 2 blocks of symmetric groups. By developing a suitable analogue of the combinatorics used by Richards, we find a formula for the corresponding “ q -decomposition numbers”, i.e. the canonical basis coefficients in the level- 1 q -deformed Fock space of type A 2 ⁢ n ( 2 ) ; a special case of a conjecture of Leclerc and Thibon asserts that these coefficients yield the spin decomposition numbers in characteristic 2 ⁢ n + 1 . Along the way, we prove some general results on q -decomposition numbers. This paper represents the first substantial progress on canonical bases in type A 2 ⁢ n ( 2 )

Simultaneous core multipartitions

We initiate the study of simultaneous core multipartitions, generalising simultaneous core partitions, which have been studied extensively in the recent literature. Given a multipartition datum (s|c), which consists of a non-negative integer s and an l-tuple c of integers, we introduce the notion of an (s|c)-core multipartition. Given an arbitrary set of multicore data, we give necessary and sufficient conditions for the corresponding set of simultaneous core multipartitions to be finite. We then study the special case of simultaneous core bipartitions, giving exact enumerative results in some special subcases.Comment: In this version the conjectures at the end of Section 4 have been extende