In the top to random shuffle, the first a cards are removed from a deck of n cards 12 · · · n and then inserted back into the deck. This action can be studied by treating the top to random shuffle as an element Ba, which we define formally in Section 2, of the algebra Q [Sn]. For a = 1, Adriano Garsia in "On the Powers of Top to Random Shuffling" (2002) derived an expansion formula for B k 1 for k ≤ n, though his proof for the formula was non-bijective. We prove, bijectively, an expansion formula for the arbitrary finite product Ba 1 Ba 2 · · · Ba k where a1, . . . , a k are positive integers, from which an improved version of Garsia's aforementioned formula follows. We show some applications of this formula for Ba 1 Ba 2 · · · Ba k , which include enumeration and calculating probabilities. Then for an arbitrary group G we define the group of G-permutations S

Parking functions were classically defined for $n$ cars attempting to park on a one-way street with $n$ parking spots, where cars only drive forward. Subsequently, parking functions have been generalized in various ways, including allowing cars the option of driving backward. The set $PF_{n,k}$ of $k$-Naples parking functions have cars who can drive backward a maximum of $k$ steps before driving forward. A recursive formula for $|PF_{n,k}|$ has been obtained, though deriving a closed formula for $|PF_{n,k}|$ appears difficult. In addition, an important subset $B_{n,k}$ of $PF_{n,k}$, called the contained $k$-Naples parking functions, has been shown, with a non-bijective proof, to have the same cardinality as that of the set $PF_n$ of classical parking functions, independent of $k$. In this paper, we study $k$-Naples parking functions in the more general context of $m$ cars and $n$ parking spots, for any $m \leq n$. We use various parking function involutions to establish a bijection between the contained $k$-Naples parking functions and the classical parking functions, from which it can be deduced that the two sets have the same number of ties. Then we extend this bijection to inject the set of $k$-Naples parking functions into a certain set of obstructed parking functions, providing an upper bound for the cardinality of the former set.

Rigged configurations are combinatorial objects prominent in the study of solvable lattice models. Marginally large tableaux are semi-standard Young tableaux of special form that give a certain realization of the crystals B(∞). We introduce cascading sequences to characterize marginally large tableaux. Then we use cascading sequences and a known nonexplicit crystal isomorphism between marginally large tableaux and rigged configurations to give a characterization of the latter set, and to give an explicit bijection between the two sets.

We introduce a new parking procedure called MVP parking in which n cars sequentially enter a one-way street with a preferred parking spot from the n parking spots on the street. If their preferred spot is empty they park there, otherwise a later car bumps a previous cars out of their preferred spot sending that car forward in the street to find the first available spot on the street. If all cars can park under this parking procedure, we say the list of preferences of the n cars is an MVP parking function of length n. We show that the set of (classical) parking functions is exactly the set of MVP parking functions although the parking outcome (order in which the cars park) is different under each parking process. Motivating the question: Given a permutation describing the outcome of the MPV parking process, what is the number of MVP parking functions resulting in that given outcome? Our main result establishes a bound for this count which is tight precisely when the permutation describing the parking outcome avoids the patterns 321 and 3412. We then consider special cases of permutations and give closed formulas for the number of MVP parking functions with those outcomes. In particular, we show that the number of MVP parking functions which park in reverse order (that is the permutation describing the outcome is the longest word in Sn, which does not avoid the pattern 321) is given by the nth Motzkin number. We also give families of permutations describing the parking outcome for which the cardinality of the set of cars parking in that order is exponential and others in which it is linear.

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