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Classical parking functions are defined as the parking preferences for n cars driving (from west to east) down a one-way street containing parking spaces labeled from 1 to n (from west to east). Cars drive down the street toward their preferred spot and park there if the spot is available. Otherwise, the car continues driving down the street and takes the first available parking space, if such a space exists. If all cars can park using this parking rule, we call the n-tuple containing the cars' parking preferences a parking function.In this paper, we introduce a generalization of the parking rule allowing cars whose preferred space is taken to first proceed up to k spaces west of their preferred spot to park before proceeding east if all of those k spaces are occupied. We call parking preferences which allow all cars to park under this new parking rule k-Naples parking functions of length n. This generalization gives a natural interpolation between classical parking functions, the case when k = 0, and all n-tuples of positive integers 1 to n, the case when k ≥ n − 1. Our main result provides a recursive formula for counting k-Naples parking functions of length n. We also give a characterization for the k = 1 case by introducing a new function that maps 1-Naples parking functions to classical parking functions, i.e. 0-Naples parking functions. Lastly, we present a bijection between k-Naples parking functions of length n whose entries are in weakly decreasing order and a family of signature Dyck paths.

Extended practice on a particular cognitive task can boost the performance of other tasks, even though they themselves have not been practiced. This transfer of benefits appears to be specific, occurring most when tasks are very similar to those being trained. But what type of similarity is most important for predicting transfer? This question is addressed with a tightly controlled randomised design, with a relatively large sample (N=175) and an adaptive control group. We created a hierarchical set of nested assessment tasks. Participants then trained on two of the tasks: one was relatively ‘low’ in the hierarchy requiring just simultaneous judgments of shapes’ spikiness, whereas the other was relatively ‘high’ requiring delayed judgments of shapes’ spikiness or number of spikes in a switching paradigm. Using the full complement of nested tasks before and after training we could then test whether and how these ‘low’ and ‘high’ training effects cascade through the hierarchy. For both training groups, relative to the control, whether or not an assessment task shared a single specific feature was the best predictor of transfer patterns. For the lower-level training group, the overall proportion of feature overlap also significantly predicted transfer, but the same was not true for the higher-level training group. Finally, pre-training between-task correlations were not predictive of the pattern of transfer for either group. Together these findings provide an experimental exploration of the specificity of transfer, and establish the nature of task overlap that is crucial for the transfer of performance improvements.

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The q-analog of Kostant's weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-analog of Kostant's partition function. This formula, when evaluated at q = 1, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra sl4(C) and give closed formulas for the q-analog of Kostant's weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the q-analog of Kostant's partition function by counting restricted colored integer partitions. These formulas, when evaluated at q = 1, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostant's weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of sl4(C), which are associated to the Weyl alternation sets. This work answers a question posed in 2019