Customer relationship management is one of the hottest management tools today. But more than half of all CRM initiatives fail to produce the anticipated results. Why? And what can companies do to reverse that negative trend? The authors--three senior Bain consultants--have spent the past ten years analyzing customer-loyalty initiatives, both successful and unsuccessful, at more than 200 companies in a wide range of industries. They've found that CRM backfires in part because executives don't understand what they are implementing, let alone how much it will cost or how long it will take. The authors' research unveiled four common pitfalls that managers stumble into when trying to implement CRM. Each pitfall is a consequence of a single flawed assumption--that CRM is software that will automatically manage customer relationships. It isn't. Rather, CRM is the creation of customer strategies and processes to build customer loyalty, which are then supported by the technology. This article looks at best practices in CRM at several companies, including the New York Times Company, Square D, GE Capital, Grand Expeditions, and BMC Software. It provides an intellectual framework for any company that wants to start a CRM program or turn around a failing one.
Offline goal-conditioned reinforcement learning methods have shown promise for reach-avoid tasks, where an agent must reach a target state while avoiding undesirable regions of the state space. Existing approaches typically encode avoid-region information into an augmented state space and cost function, which prevents flexible, dynamic specification of novel avoid-region information at evaluation time. They also rely heavily on well-designed reward and cost functions, limiting scalability to complex or poorly structured environments. We introduce RADT, a decision transformer model for offline, reward-free, goal-conditioned, avoid region-conditioned RL. RADT encodes goals and avoid regions directly as prompt tokens, allowing any number of avoid regions of arbitrary size to be specified at evaluation time. Using only suboptimal offline trajectories from a random policy, RADT learns reach-avoid behavior through a novel combination of goal and avoid-region hindsight relabeling. We benchmark RADT against 3 existing offline goal-conditioned RL models across 11 tasks, environments, and experimental settings. RADT generalizes in a zero-shot manner to out-of-distribution avoid region sizes
A word contains a \emph{half-flip} if it contains non-empty factors $uv$ and $vu$ where $|u|=|v|$. Fici reports a non-constructive proof of the existence of an infinite word over a finite alphabet avoiding half-flips and asks for the size of the smallest alphabet over which half-flips may be avoided. Currie and Rampersad have proposed a pure morphic word over 8 letters and a morphic word over 5 letters and conjecture that they avoid half-flips. We present a pure morphic word over 5 letters that avoids half-flips. We also show that half-flips with $|u|\ge2$ are 3-avoidable and that half-flips with $|u|\ge4$ are 2-avoidable.
A rotation of a permutation is a new permutation obtained by moving the first several terms of the permutation to the end of the permutation. A circular permutation is the set of all rotations of a permutation. The enumerations of permutations and circular permutations avoiding patterns of length three and four are well studied. In this paper, we provide exact formulas for the number of permutations whose first $k\geq 2$ rotations all avoid a given pattern of length three, as well as the number of permutations whose first three rotations respectively avoid the rotations of a given pattern of length three. In contrast to permutations and circular permutations avoiding patterns of length three, the Wilf-equivalence classes under study are entirely determined by complements and reverses. We also classify and enumerate permutations whose first two rotations avoid different patterns of length three.
In 2019, Bóna and Smith introduced the notion of strong pattern avoidance, saying that a permutation $π$ strongly avoids a pattern $σ$ if $π$ and $π^2$ both avoid $σ$. Recently, Archer and Geary generalized the idea of strong pattern avoidance to chain avoidance, in which a permutation $π$ avoids a chain of patterns $(τ^{(1)}:τ^{(2)}:\cdots:τ^{(k)})$ if the $i$-th power of the permutation avoids the pattern $τ^{(i)}$ for $1\leq i\leq k$. In this paper, we give explicit formulae for the number of sets of permutations avoiding certain chains of patterns. Our results give affirmative answers to two conjectures proposed by Archer and Geary.
Non-crossing and non-nesting permutations are variations of the well-known Stirling permutations. A permutation $π$ on $\{1,1,2,2,\ldots, n,n\}$ is called non-crossing if it avoids the crossing patterns $\{1212,2121\}$ and is called non-nesting if it avoids the nesting patterns $\{1221,2112\}.$ Pattern avoidance in these permutations has been considered in recent years, but it has remained open to enumerate the non-crossing and non-nesting permutations that avoid a single pattern of length 3. In this paper, we provide generating functions for those non-crossing and non-nesting permutations that avoid the pattern 231 (and, by symmetry, the patterns 132, 213, or 312).
A permutation $π$ is said to avoid a chain $(σ:τ)$ of patterns if $π$ avoids $σ$ and $π^2$ avoids $τ.$ In this paper, we define a notion of pattern avoidance for compositions of positive integers and use that idea to enumerate permutations of length $n$ that avoid the chain $(312,321:σ)$ for any pattern $σ\in \bigcup_{m\geq 1} S_m$. We also enumerate those permutations that avoid the chain $(312,4321:σ)$ for any $σ\in S_3.$
In a recent paper, Bona and Smith define the notion of \textit{strong avoidance}, in which a permutation and its square both avoid a given pattern. In this paper, we generalize this idea to what we call \textit{chain avoidance}. We say that a permutation avoids a chain of patterns $(τ_1 : τ_2: \cdots : τ_k)$ if the $i$-th power of the permutation avoids the pattern $τ_i$. We enumerate the set of permutations $π$ which avoid the chain $(213, 312 : τ)$, i.e.,~unimodal permutations whose square avoids $τ$, for $τ\in §_3$ and use this to find a lower bound on the number of permutations that avoid the chain $(312: τ)$ for $τ\in §_3$. We finish the paper by discussing permutations that avoid longer chains.
A permutation $π$ strongly avoids the pattern $τ$ if both $π$ and $π^2$ avoid $τ$. In this paper, we enumerate permutations of size $n$ that strongly avoid the pattern 132. This enumeration allows us to prove a conjecture that the growth rate of such permutations is 2.
We show that cyclic permutations avoiding $321$ are precisely those permutations whose image under the fundamental bijection avoid a set of vincular patterns. We do this by using pattern functions and arrow patterns, in combination with the characterization of $321$ avoidance in terms of equality of the upper bound of the Daiconis-Graham inequalities. We then explore some consequences of this result, including upper and lower bound results on the growth rate of $321$ avoiding cycles.
Shallow permutations were defined in 1977 to be those that satisfy the lower bound of the Diaconis-Graham inequality. Recently, there has been renewed interest in these permutations. In particular, Berman and Tenner showed they satisfy certain pattern avoidance conditions in their cycle form and Woo showed they are exactly those whose cycle diagrams are unlinked. Shallow permutations that avoid 321 have appeared in many contexts; they are those permutations for which depth equals the reflection length, they have unimodal cycles, and they have been called Boolean permutations. Motivated by this interest in 321-avoiding shallow permutations, we investigate $σ$-avoiding shallow permutations for all $σ\in \mathcal{S}_3$. To do this, we develop more general structural results about shallow permutations, and apply them to enumerate shallow permutations avoiding any pattern of length 3.
Nonnesting permutations are permutations of the multiset $\{1,1,2,2,\dots,n,n\}$ that avoid subsequences of the form $abba$ for any $a eq b$. These permutations have recently been studied in connection to noncrossing (also called quasi-Stirling) permutations, which are those that avoid subsequences of the form $abab$, and in turn generalize the well-known Stirling permutations. Inspired by the work by Archer et al. on pattern avoidance in noncrossing permutations, we consider the analogous problem in the nonnesting case. We enumerate nonnesting permutations that avoid each set of two or more patterns of length 3, as well as those that avoid some sets of patterns of length 4. We obtain closed formulas and generating functions, some of which involve unexpected appearances of the Catalan and Fibonacci numbers. Our proofs rely on decompositions, recurrences, and bijections.
Discriminative approaches to classification often learn shortcuts that hold in-distribution but fail even under minor distribution shift. This failure mode stems from an overreliance on features that are spuriously correlated with the label. We show that generative classifiers, which use class-conditional generative models, can avoid this issue by modeling all features, both core and spurious, instead of mainly spurious ones. These generative classifiers are simple to train, avoiding the need for specialized augmentations, strong regularization, extra hyperparameters, or knowledge of the specific spurious correlations to avoid. We find that diffusion-based and autoregressive generative classifiers achieve state-of-the-art performance on five standard image and text distribution shift benchmarks and reduce the impact of spurious correlations in realistic applications, such as medical or satellite datasets. Finally, we carefully analyze a Gaussian toy setting to understand the inductive biases of generative classifiers, as well as the data properties that determine when generative classifiers outperform discriminative ones.
Hamilton-Jacobi (HJ) reachability analysis is a fundamental tool for the safety verification and control synthesis of nonlinear control systems. Classical HJ reachability analysis methods compute value functions over grids which discretize the continuous state space. Such approaches do not account for discretization errors and thus do not guarantee that the sets represented by the computed value functions over-approximate the backward reachable sets (BRS) when given avoid specifications or under-approximate the reach-avoid sets (RAS) when given reach-avoid specifications. We address this issue by presenting an algorithm for computing sound upper and lower bounds on the HJ value functions that guarantee the sound over-approximation of BRS and under-approximation of RAS. Additionally, we develop a refinement algorithm that splits the grid cells which could not be classified as within or outside the BRS or RAS given the computed bounds to obtain corresponding tighter bounds. We validate the effectiveness of our algorithm in two case studies.
Avoidance games are games in which two players claim vertices of a hypergraph and try to avoid some structures. These games are studied since the introduction of the game of SIM in 1968, but only few complexity results are known on them. In 2001, Slany proved some partial results on Avoider-Avoider games complexity, and in 2017 Bonnet et al. proved that short Avoider-Enforcer games are Co-W[1]-hard. More recently, in 2022, Miltzow and Stojaković proved that these games are NP-hard. As these games corresponds to the misère version of the well-known Maker-Breaker games, introduced in 1963 and proven PSPACE-complete in 1978, one could expect these games to be PSPACE-complete too, but the question remained open since then. We prove here that both Avoider-Avoider and Avoider-Enforcer conventions are PSPACE-complete, and as a consequence of it that some particular Avoider-Enforcer games also are.
Let $π$ be a cycle permutation that can be expressed as one-line $π= π_1π_2 \cdot\cdot\cdot π_n$ and a cycle form $π= (c_1,c_2, ..., c_n)$. Archer et al. introduced the notion of pattern avoidance of one-line and all cycle forms for a cycle permutation $π$, defined as $π_1π_2 \cdot\cdot\cdot π_n$ and its arbitrary cycle form $c_ic_{i+1}\cdot\cdot\cdot c_nc_1c_2\cdot\cdot\cdot c_{i-1}$ avoid a given pattern. Let $\mathcal{A}^\circ_n(σ; τ)$ denote the set of cyclic permutations in the symmetric group $S_n$ that avoid $σ$ in their one-line form and avoid $τ$ in their all cycle forms. In this note, we prove that $|\mathcal{A}^\circ_n(2431; 1324)|$ is the $(n-1)^{\rm{st}}$ Pell number for any positive integer $n$. Thereby, we give a positive answer to a conjecture of Archer et al.
The fundamental bijection is a bijection $θ:\mathcal{S}_n\to\mathcal{S}_n$ in which one uses the standard cycle form of one permutation to obtain another permutation in one-line form. In this paper, we enumerate the set of permutations $π\in \mathcal{S}_n$ that avoids a pattern $σ\in \mathcal{S}_3$, whose image $θ(π)$ also avoids $σ$. We additionally consider what happens under repeated iterations of $θ$; in particular, we enumerate permutations $π\in \mathcal{S}_n$ that have the property that $π$ and its first $k$ iterations under $θ$ all avoid a pattern $σ$. Finally, we consider permutations with the property that $π=θ^2(π)$ that avoid a given pattern $σ$, and end the paper with some directions for future study.
In 2019, Bóna and Smith introduced the notion of \emph{strong pattern avoidance}, that is, a permutation and its square both avoid a given pattern. In this paper, we enumerate the set of permutations $π$ which not only strongly avoid the pattern $312$ or $231$ but also avoid the pattern $τ$, for $τ\in S_3$ and some $τ\in S_4$. One of them is to give a positive answer to a conjecture of Archer and Geary.
A permutation of the integers avoiding monotone arithmetic progressions of length $6$ was constructed in (Geneson, 2018). We improve on this by constructing a permutation of the integers avoiding monotone arithmetic progressions of length $5$. We also construct permutations of the integers and the positive integers that improve on previous upper and lower density results. In (Davis et al. 1977) they constructed a doubly infinite permutation of the positive integers that avoids monotone arithmetic progressions of length $4$. We construct a doubly infinite permutation of the integers avoiding monotone arithmetic progressions of length $5$. A permutation of the positive integers that avoided monotone arithmetic progressions of length $4$ with odd common difference was constructed in (LeSaulnier and Vijay, 2011). We generalise this result and show that for each $k\geq 1$, there exists a permutation of the positive integers that avoids monotone arithmetic progressions of length $4$ with common difference not divisible by $2^k$. In addition, we specify the structure of permutations of $[1,n]$ that avoid length $3$ monotone arithmetic progressions mod $n$ as defined in (Davis et al. 1977)
How many matchings on the vertex set V={1,2,...,2n} avoid a given configuration of three edges? Chen, Deng and Du have shown that the number of matchings that avoid three nesting edges is equal to the number of matchings avoiding three pairwise crossing edges. In this paper, we consider other forbidden configurations of size three. We present a bijection between matchings avoiding three crossing edges and matchings avoiding an edge nested below two crossing edges. This bijection uses non-crossing pairs of Dyck paths of length 2n as an intermediate step. Apart from that, we give a bijection that maps matchings avoiding two nested edges crossed by a third edge onto the matchings avoiding all configurations from an infinite family, which contains the configuration consisting of three crossing edges. We use this bijection to show that for matchings of size n>3, it is easier to avoid three crossing edges than to avoid two nested edges crossed by a third edge. In this updated version of this paper, we add new references to papers that have obtained analogous results in a different context.