Cyclic sieving of increasing tableaux and small Schröder paths
نویسنده
چکیده
An increasing tableau is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a bijection between rectangular 2-row increasing tableaux and small Schröder paths. Using the jeu de taquin for increasing tableaux of [Thomas–Yong ’09], we then present a new instance of the cyclic sieving phenomenon of [Reiner–Stanton–White ’04]. Résumé. Un tableau croissant est un tableau semi-standard avec les lignes et les colonnes croissantes au sens strict. Il est bien connu que les nombres de Catalan énumèrent les tableaux de Young standard rectangulaires de deux lignes et aussi les chemins de Dyck. Nous généralisons ceci pour une bijection entre tableaux croissants rectangulaires á 2 lignes et petits chemins de Schröder. Utilisant le jeu de taquin de [Thomas–Yong ’09] pour tableaux croissants, nous présentons ensuite une nouvelle instance du phénomène du crible cyclique de [Reiner–Stanton–White ’04].
منابع مشابه
Promotion and Cyclic Sieving via Webs
We show that Schützenberger’s promotion on two and three row rectangular Young tableaux can be realized as cyclic rotation of certain planar graphs introduced by Kuperberg. Moreover, following work of the third author, we show that this action admits the cyclic sieving phenomenon.
متن کاملJeu-de-taquin promotion and a cyclic sieving phenomenon for semistandard hook tableaux
Jeu-de-taquin promotion yields a bijection on the set of semistandard λ-tableaux with entries bounded by k. In this note, we determine the order of jeu-de-taquin promotion on the set of semistandard hook tableaux CST ((n−m, 1m), k), with entries bounded by k, and on the set of semistandard hook tableaux with fixed content α, CST ((n−m, 1m), k, α). We give a bijection between CST ((n−m, 1m), k, ...
متن کاملInvariant Tensors and the Cyclic Sieving Phenomenon
We construct a large class of examples of the cyclic sieving phenomenon by exploiting the representation theory of semi-simple Lie algebras. Let M be a finite dimensional representation of a semi-simple Lie algebra and let B be the associated Kashiwara crystal. For r > 0, the triple (X, c, P ) which exhibits the cyclic sieving phenomenon is constructed as follows: the set X is the set of isolat...
متن کاملCyclic sieving, promotion, and representation theory
We prove a collection of conjectures of D. White [37], as well as some related conjectures of Abuzzahab-Korson-Li-Meyer [1] and of Reiner and White [21], [37], regarding the cyclic sieving phenomenon of Reiner, Stanton, and White [22] as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of ...
متن کاملMaximal Fillings of Moon Polyominoes, Simplicial Complexes, and Schubert Polynomials
We exhibit a canonical connection between maximal (0, 1)-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable, and thus shellable, sphere. In particular, this implies a positivity result for Schubert polynomials. Mor...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Comb. Theory, Ser. A
دوره 125 شماره
صفحات -
تاریخ انتشار 2014