If we compose sufficiently many random functions on a finite set, then the composite function will be constant. We determine the number of compositions that are needed, on average. Choose random functions $f_1, f_2,f_3,\dots $ independently and uniformly from among the $n^n$ functions from $[n]$ into $[n]$. For $t>1$, let $g_t=f_t\circ f_{t-1}\circ \cdots \circ f_1$ be the composition of the first $t$ functions. Let $T$ be the smallest $t$ for which $g_t$ is constant(i.e. $g_t(i)=g_t(j)$ for all $i,j$). We prove that $E(T)\sim 2n$ as $n\rightarrow\infty$, where $E(T)$ denotes the expected value of $T$.

International audience We introduce two families of symmetric functions with an extra parameter $t$ that specialize to Schubert representatives for cohomology and homology of the affine Grassmannian when $t=1$. The families are defined by a statistic on combinatorial objects associated to the type-$A$ affine Weyl group and their transition matrix with Hall-Littlewood polynomials is $t$-positive. We conjecture that one family is the set of $k$-atoms. Nous présentons deux familles de fonctions symétriques dépendant d'un paramètre $t$ et dont les spécialisations à $t=1$ correspondent aux classes de Schubert dans la cohomologie et l'homologie des variétés Grassmanniennes affines. Les familles sont définies par des statistiques sur certains objets combinatoires associés au groupe de Weyl affine de type $A$ et leurs matrices de transition dans la base des polynômes de Hall-Littlewood sont $t$-positives. Nous conjecturons qu'une de ces familles correspond aux $k$-atomes.

International audience We give a new description of the Pieri rule for $k$-Schur functions using the Bruhat order on the affine type-$A$ Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and $k$-Schur functions. Nous présentons une nouvelle description, issue de l'ordre de Bruhat du groupe de Weyl affine de type $A$, de la règle de Pieri pour les fonctions $k$-Schur. Ce faisant, nous obtenons une nouvelle formule combinatoire pour les représentants des classes de Schubert de la cohomologie des Grassmannienne affines. Nous décrivons aussi comment notre approche permet d'obtenir les polynômes de Kostka-Foulkes et comment elle peut être appliquée à l’étude des matrices de transition entre les polynômes de Hall-Littlewood et les fonctions $k$-Schur.

We definitively establish that the theory of symmetric Macdonald polynomials aligns with quantum and affine Schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong (Bruhat) order poset on the type-A affine Weyl group. We construct two one-parameter families of functions that respectively transition positively with Hall-Littlewood and Macdonald's P-functions, and specialize to the representatives for Schubert classes of homology and cohomology of the affine Grassmannian. Our approach leads us to conjecture that all elements in a defining set of 3-point genus 0 Gromov-Witten invariants for flag manifolds can be formulated as strong covers.Assuming this conjecture, since A (k) λ (x; t) is a t-positive sum of Schur functions, the Macdonald/Schur transition matrices factor over N [q, t]. The construction of A (k) µ (x; t) is extremely intricate and the conjectures remain unproven as a consequence. Nevertheless, their study inspired discoveries in representation theory [Hai08] and suggested a connection between the theory of Macdonald polynomials and quantum and affine Schubert calculus.The affine Grassmannian of G = S L(n, C) is given by Gr] is the ring of formal power series and C((t)) = C[[t]][t −1 ] is the ring of formal Laurent series. Quillen (unpublished) and Garland and Raghunathan [GR75] showed that Gr is homotopy-equivalent to the group Ω SU(n, C) of based loops into SU(n, C). The homology H * (Gr) and cohomology H * (Gr) thus have dual Hopf algebra structures which, using results of [Bot58], can be explicitly identified by a subring Λ (n) and a quotient Λ (n) of Λ.On one hand, the algebraic nil-Hecke ring construction of Kostant and Kumar [KK86] and the work of Peterson [Pet97] developed the study of Schubert bases associated to Schubert cells of Gr in the Bruhat decomposition of G (C((t))),indexed by Grassmannian elements of the affine Weyl groupà n−1 . On the other, inspired by an empirical study of the polynomials A (k) λ (x; t) when t = 1, distinguished bases for Λ (n) and Λ (n) that refine the Schur basis for Λ were introduced and connected to the quantum cohomology of Grassmannians in [LM08, LM07]. The two approaches merged when Lam proved in [Lam08] that these k-Schur bases are sets of representatives for the Schubert classes of H * (Gr) and H * (Gr) (where k = n − 1). Moreover, the k-Schur functions for Λ (k) = Λ t=1 (k) were conjectured [LM05] to be the parameterless {A (k) λ (x; 1)}, suggesting a link from the theory of Macdonald polynomials to quantum and affine Schubert calculus.Here, we circumvent the problem that the characterization for A (k) µ (x; t) lacks in mechanism for proof and definitively establish this link. Our work relies on a remarkable connection between chains in the strong and the weak order poset on the type-A affine Weyl group. From this, we are able to construct one parameter families of symmetric functions that transition positively with H µ (x; 0, t) and Macdonald's P-functions and that specialize to the Schubert repres...

We give a new description of the Pieri rule for k-Schur functions using the Bruhat order on the affine type-A Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and k-Schur functions.Nous présentons une nouvelle description, issue de l'ordre de Bruhat du groupe de Weyl affine de type A, de la règle de Pieri pour les fonctions k-Schur. Ce faisant, nous obtenons une nouvelle formule combinatoire pour les représentants des classes de Schubert de la cohomologie des Grassmannienne affines. Nous décrivons aussi comment notre approche permet d'obtenir les polynômes de Kostka-Foulkes et comment elle peut être appliquée à l'étude des matrices de transition entre les polynômes de Hall-Littlewood et les fonctions k-Schur.