Note: Random-to-front shuffles on trees

A Markov chain is considered whose states are orderings of an underlying fixed tree and whose transitions are local"random-to-front"reorderings, driven by a probability distribution on subsets of the leaves. The eigenvalues of the transition matrix are determined using Brown's theory of random walk on semigroups.


Introduction
The random-to-front shuffle of a linear list (known in the card-game model also as "inverse riffle shuffle") is a well-known and much studied finite-state Markov chain. Its states are the linear orderings of an underlying finite set, and a step of the chain results from selecting a subset (often a singleton) and moving it to the front of the current list in the induced order. See e.g. [2,5,7] and the references given there. In this note we consider a slight generalization, namely to shuffles on trees.
Consider a fixed rooted tree T whose leaves L are all at the same depth. The following shows a such a tree of depth 3. Suppose that at each inner node (i.e., node that is not a leaf) a total ordering of its children is given. For instance, it can be the left-to-right ordering given by a planar drawing of the tree, such as in Figure 1. Now, a subset E of the set of leaves L is chosen with some probability. Then the ordering is rearranged locally at each inner node so that the children having some descendant in E come first, and otherwise the induced order is preserved. The process is illustrated in Figures 3 and 4.
In this note the eigenvalues of the transition matrix of this Markov chain are determined. This is a straight-forward application of Brown's theory of random walks on semigroups [4].
Note that if depth(T ) = 1 the Markov chain we describe amounts to the classical linear random-to-front shuffle. For depth(T ) > 1 we perform such a linear shuffle locally at each inner node, in each case moving the set of E-related nodes to the front. If depth(T ) = 2 we obtain the "library with several shelves" model considered in [3], as indicated in Figure 2. This case was derived in [3] via geometric considerations, ultimately relying on Brown's theory of random walks on semigroups.
If one cares only about the library result, and not about random walks on complex hyperplane arrangements, there is of course no need to mix in geometric considerations. This note can be seen as a self-contained appendix to [3] whose modest purpose is to fill in the details on how to obtain the general dynamic library model in the simplest and most direct way, avoiding geometry.
Another "tree analogue" of the classical linear random-to-front shuffle, different from the one considered here, has been studied in the literature. This is the random-to-root shuffle on binary trees, see e.g. [1,6].

Shuffles on trees
We begin by establishing notation. For any finite set A, let The sets Π(A) and Π ord (A) are partially ordered by refinement, meaning that α ≤ β if and only if every block of the partition (or ordered partition) α is a union of blocks from β. Direct products (of sets, posets, . . . ) are denoted by .
We consider rooted trees T that are pure, meaning that all leaves are at the same depth d. Let V j denote the set of nodes at depth j. So, For each inner node x ∈ I, let C x denote the set of its children. The subsets of L act on O(T ) in the following way.
Definition 2.3. Let π = (π x ) x∈I be a local ordering, and let E ∈ 2 L . Then is the linear ordering of C x in which the Erelated elements come first, in the order induced by π x , followed by the remaining elements, also in the induced order.
The following figure shows a local ordering π of a tree T , which coincides with left-to-right order in the planar drawing of T . The indicated choice E of leaves induces a move to the following local ordering E(π). The E-related nodes are shaded. Let Part(T ) def = x∈I Π(C x ). So, an element α ∈ Part(T ) is a choice of partition α x of the set of children of x, for each inner node x. The following special elements of Part(T ) are induced by subsets E ⊆ L. For each x ∈ I let α E x be the partition of C x into two blocks, one block consisting of the E-related elements and one of the remaining elements (one of these blocks may be empty, in which case we forget it).
x for every x ∈ I. Notice that for every nontrivial α ∈ Part(T ) there exists some α-compatible proper subset E ⊆ L.
Theorem 2.6. Let T be a pure tree with leaves L. Furthermore, let {w E } E⊆L be a probability distribution on 2 L and P w the transition matrix of the induced random walk on local orderings of T : (iii) The multiplicity of the eigenvalue ε α is These are all the eigenvalues of P w .
Proof. As mentioned in the introduction, this is a special case of Brown's theory for walks on semigroups [4], with which we now assume familiarity. Let Part ord (T ) def = x∈I Π ord (C x ). So, an element β ∈ Part ord (T ) is a choice of ordered partition β x of the set of children of x, for each inner node x. In particular, for each subset E ⊆ L there is an element β E ∈ Part ord (T ) whose component β E x at x ∈ I is the two-block ordered partition of C x whose first block consists of the E-related elements of C x , and second block of the remainder. (If one of these blocks is empty we forget about it and let β E x have only one block.) Now, introduce the following probability distribution on Part ord (T ): for all other ordered partitions.
Given this set-up, the proof consists of verifying each of the following claims for Part ord (T ), and then referring to [4].
(1) Part ord (T ) is an LRB (left regular band) semigroup with component-wise composition. The composition in each factor Π ord (A) has the following description. If X = X 1 , . . . , X p and Y = Y 1 , . . . , Y q are ordered partitions of A, then X • Y = X i ∩ Y j with the blocks ordered by the lexicographic order of the pairs of indices (i, j). (2) Its support lattice is Part(T ) and support map supp : Part ord (T ) → Part(T ), whose component at each x ∈ I is the map Π ord (C x ) → Π(C x ) that sends an ordered partition of C x to an unordered partition by forgetting the ordering of its blocks.  [4] by the probability assignment (2.1), are precisely the steps described in Definition 2.4. (5) For each E ⊆ L and α ∈ Part(T ): |µ(α, 1)| computed on the product partition lattice Part(T ). From this follows, via Brown's theory [4], that for all α ∈ Part(T ). By the product property of the Möbius function and its well-known explicit evaluation on the partition lattice (see [8]), this quantity equals In view of these facts the theorem is obtained by specializing Theorem 1 on page 880 of [4] to the semigroup Part ord (T ).

Remarks
3.1. The random walk of Theorem 2.6 has a unique stationary distribution π if and only if {E ∈ 2 L : w E > 0} is separating, meaning that for every inner node x ∈ I and every pair of siblings y, z ∈ C x , y = z, there is a subset E ⊆ L with w E > 0 for which one of y and z is E-related and the other is not. This follows from Theorem 2 of Brown and Diaconis [5], using the fact that the random walk we consider can be realized as a walk on the complement of a product of real braid arrangements. Theorem 2 of [5] also gives additional information about the stationary distribution.
3.2. One easily checks that the subset {β E : E ⊆ L} generates the full semigroup Part ord (T ), and that the set of its maximal elements O(T ) is generated by {β {e} : e ∈ L}.
3.3. Suppose that w E = 0 only if |E| = 1. Then Theorem 2.6 implies that the eigenvalues are indexed by x∈I 2 Cx , and that their multiplicities are products of derangement numbers, thus generalizing the well-known result of Donnelly, Kapoor-Reingold and Phatarfod for the Tsetlin library (the depth(T ) = 1 case); see the references for this given in [2,4,5].