Generating functions for Wilf equivalence under generalized factor order

Kitaev, Liese, Remmel, and Sagan recently defined generalized factor order on words comprised of letters from a partially ordered set $(P, \leq_P)$ by setting $u \leq_P w$ if there is a subword $v$ of $w$ of the same length as $u$ such that the $i$-th character of $v$ is greater than or equal to the $i$-th character of $u$ for all $i$. This subword $v$ is called an embedding of $u$ into $w$. For the case where $P$ is the positive integers with the usual ordering, they defined the weight of a word $w = w_1\ldots w_n$ to be $\text{wt}(w) = x^{\sum_{i=1}^n w_i} t^{n}$, and the corresponding weight generating function $F(u;t,x) = \sum_{w \geq_P u} \text{wt}(w)$. They then defined two words $u$ and $v$ to be Wilf equivalent, denoted $u \backsim v$, if and only if $F(u;t,x) = F(v;t,x)$. They also defined the related generating function $S(u;t,x) = \sum_{w \in \mathcal{S}(u)} \text{wt}(w)$ where $\mathcal{S}(u)$ is the set of all words $w$ such that the only embedding of $u$ into $w$ is a suffix of $w$, and showed that $u \backsim v$ if and only if $S(u;t,x) = S(v;t,x)$. We continue this study by giving an explicit formula for $S(u;t,x)$ if $u$ factors into a weakly increasing word followed by a weakly decreasing word. We use this formula as an aid to classify Wilf equivalence for all words of length 3. We also show that coefficients of related generating functions are well-known sequences in several special cases. Finally, we discuss a conjecture that if $u \backsim v$ then $u$ and $v$ must be rearrangements, and the stronger conjecture that there also must be a weight-preserving bijection $f: \mathcal{S}(u) \rightarrow \mathcal{S}(v)$ such that $f(u)$ is a rearrangement of $u$ for all $u$.Comment: 23 page

Enumeration of super-strong Wilf equivalence classes of permutations in the generalized factor order

Super-strong Wilf equivalence classes of the symmetric group ${\mathcal S}_n$ on $n$ letters, with respect to the generalized factor order, were shown by Hadjiloucas, Michos and Savvidou (2018) to be in bijection with pyramidal sequences of consecutive differences. In this article we enumerate the latter by giving recursive formulae in terms of a two-dimensional analogue of non-interval permutations. As a by-product, we obtain a recursively defined set of representatives of super-strong Wilf equivalence classes in ${\mathcal S}_n$. We also provide a connection between super-strong Wilf equivalence and the geometric notion of shift equivalence---originally defined by Fidler, Glasscock, Miceli, Pantone, and Xu (2018) for words---by showing that an alternate way to characterize super-strong Wilf equivalence for permutations is by keeping only rigid shifts in the definition of shift equivalence. This allows us to fully describe shift equivalence classes for permutations of size $n$ and enumerate them, answering the corresponding problem posed by Fidler, Glasscock, Miceli, Pantone, and Xu (2018).Comment: 18 pages, 5 table

Shift Equivalence in the Generalized Factor Order

We provide a geometric condition that guarantees strong Wilf equivalence in the generalized factor order. This provides a powerful tool for proving specific and general Wilf equivalence results, and several such examples are given

Generating Functions and Wilf Equivalence for Generalized Interval Embeddings

In 1999 in [J. Difference Equ. Appl. 5, 355–377], Noonan and Zeilberger extended the Goulden-Jackson Cluster Method to find generating functions of word factors. Then in 2009 in [Electron. J. Combin. 16(2), RZZ], Kitaev, Liese, Remmel and Sagan found generating functions for word embeddings and proved several results on Wilf-equivalence in that setting. In this article, the authors focus on generalized interval embeddings, which encapsulate both factors and embeddings, as well as the “space between” these two ideas. The authors present some results in the most general case of interval embeddings. Two special cases of interval embeddings are also discussed, as well as their relationship to results in previous works in the area of pattern avoidance in words

On super-strong Wilf equivalence classes of permutations

Super-strong Wilf equivalence is a type of Wilf equivalence on words that was originally introduced as strong Wilf equivalence by Kitaev et al. [Electron. J. Combin. 16(2)] in 2009. We provide a necessary and sufficient condition for two permutations in n letters to be super-strongly Wilf equivalent, using distances between letters within a permutation. Furthermore, we give a characterization of such equivalence classes via two-colored binary trees. This allows us to prove, in the case of super-strong Wilf equivalence, the conjecture stated in the same article by Kitaev et al. that the cardinality of each Wilf equivalence class is a power of 2

The combinatorics of Jeff Remmel

We give a brief overview of the life and combinatorics of Jeff Remmel, a mathematician with successful careers in both logic and combinatorics

Algorithms Seminar, 2001-2002

These seminar notes constitute the proceedings of a seminar devoted to the analysis of algorithms and related topics. The subjects covered include combinatorics, symbolic computation, asymptotic analysis, number theory, as well as the analysis of algorithms, data structures, and network protocols