scholarly journals The Rearrangement Conjecture

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jay Pantone ◽  
Vincent Vatter
Keyword(s):  
International Audience ◽  
Wilf Equivalence ◽  
Factor Order

International audience The Rearrangement Conjecture states that if two words over $\mathbb{P}$ are Wilf-equivalent in the factor order on $\mathbb{P}^{\ast}$ then they are rearrangements of each other. We introduce the notion of strong Wilf-equivalence and prove that if two words over $\mathbb{P}$ are strongly Wilf-equivalent then they are rearrangements of each other. We further conjecture that Wilf-equivalence implies strong Wilf-equivalence.

scholarly journals Rationality, irrationality, and Wilf equivalence in generalized factor order

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Sergey Kitaev ◽  
Jeffrey Liese ◽  
Jeffrey Remmel ◽  
Bruce Sagan
Keyword(s):  
Generating Function ◽  
Free Monoid ◽  
Recursive Formula ◽  
Finite State Automaton ◽  
Finite State ◽  
Pumping Lemma ◽  
International Audience ◽  
Wilf Equivalence ◽  
Positive Integers ◽  
Factor Order

International audience Let $P$ be a partially ordered set and consider the free monoid $P^{\ast}$ of all words over $P$. If $w,w' \in P^{\ast}$ then $w'$ is a factor of $w$ if there are words $u,v$ with $w=uw'v$. Define generalized factor order on $P^{\ast}$ by letting $u \leq w$ if there is a factor $w'$ of $w$ having the same length as $u$ such that $u \leq w'$, where the comparison of $u$ and $w'$ is done componentwise using the partial order in $P$. One obtains ordinary factor order by insisting that $u=w'$ or, equivalently, by taking $P$ to be an antichain. Given $u \in P^{\ast}$, we prove that the language $\mathcal{F}(u)=\{w : w \geq u\}$ is accepted by a finite state automaton. If $P$ is finite then it follows that the generating function $F(u)=\sum_{w \geq u} w$ is rational. This is an analogue of a theorem of Björner and Sagan for generalized subword order. We also consider $P=\mathbb{P}$, the positive integers with the usual total order, so that $\mathbb{P}^{\ast}$ is the set of compositions. In this case one obtains a weight generating function $F(u;t,x)$ by substituting $tx^n$ each time $n \in \mathbb{P}$ appears in $F(u)$. We show that this generating function is also rational by using the transfer-matrix method. Words $u,v$ are said to be Wilf equivalent if $F(u;t,x)=F(v;t,x)$ and we can prove various Wilf equivalences combinatorially. Björner found a recursive formula for the Möbius function of ordinary factor order on $P^{\ast}$. It follows that one always has $\mu (u,w)=0, \pm 1$. Using the Pumping Lemma we show that the generating function $M(u)= \sum_{w \geq u} | \mu (u,w) | w$ can be irrational. Soit $P$ un ensemble partiellement ordonné. Nous considérons le monoïde libre $P^{\ast}$ de tous les mots utilisant $P$ comme alphabet. Si $w,w' \in P^{\ast}$, on dit que $w'$ est un facteur de $w$ s'il y a des mots $u,v$ avec $w=uw'v$. Nous définissons l'ordre facteur généralisé sur $P^{\ast}$ par: $u \leq w$ s'il y a un facteur $w'$ de $w$ ayant la même longueur que $u$ tel que $u \leq w'$, où la comparaison de $u$ avec $w'$ est faite lettre par lettre utilisant l'ordre en $P$. On obtient l'ordre facteur usuel si on insiste que $u=w'$ ou, ce qui est la même chose, en prenant $P$ comme antichaîne. Pour n'importe quel $u \in P^{\ast}$, nous démontrons que le langage $\mathcal{F}(u)=\{w : w \geq u\}$ est accepté par un automaton avec un nombre fini d'états. Si $P$ est fini, ça implique que la fonction génératrice $F(u)=\sum_{w \geq u} w$ est rationnelle. Björner et Sagan ont démontré le théorème analogue pour l'ordre où, en la définition au-dessus, $w'$ est un sous-mot de $w$. Nous considérons aussi le cas $P=\mathbb{P}$, les entiers positifs avec l'ordre usuel, donc $P^{\ast}$ est l'ensemble des compositions. En ce cas on obtient une fonction génératrice pondéré $F(u;t,x)$ en remplaçant $tx^n$ chaque fois on trouve $n \in \mathbb{P}$ en $F(u)$. Nous démontrons que cette fonction génératrice est aussi rationnelle en utilisant la Méthode Matrice de Tranfert. On dit que let mots $u,v$ sont Wilf-équivalents si $F(u;t,x)=F(v;t,x)$. Nous pouvons démontré quelques équivalences dans une manière combinatoire. Björner a trouvé une formule récursive pour la fonction Möbius de l'ordre facteur usuel sur $P^{\ast}$. Cette formule implique qu'on a toujours $\mu (u,w)=0, \pm 1$. En utilisant le Lemme de Pompage, nous démontrons que la fonction génératrice $M(u)= \sum_{w \geq u} | \mu (u,w) | w$ peut être irrationnelle.

scholarly journals Modified Growth Diagrams, Permutation Pivots, and the BWX Map $\phi^*$

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Jonathan Bloom ◽  
Dan Saracino
Keyword(s):  
Present Article ◽  
Iterative Process ◽  
Nous Proposons ◽  
Rook Placements ◽  
Growth Diagram ◽  
International Audience ◽  
Wilf Equivalence ◽  
Ferrers Boards

International audience In their paper on Wilf-equivalence for singleton classes, Backelin, West, and Xin introduced a transformation $\phi^*$, defined by an iterative process and operating on (all) full rook placements on Ferrers boards. Bousquet-Mélou and Steingrimsson proved the analogue of the main result of Backelin, West, and Xin in the context of involutions, and in so doing they needed to prove that $\phi^*$ commutes with the operation of taking inverses. The proof of this commutation result was long and difficult, and Bousquet-Mélou and Steingrimsson asked if $\phi^*$ might be reformulated in such a way as to make this result obvious. In the present paper we provide such a reformulation of $\phi^*$, by modifying the growth diagram algorithm of Fomin. This also answers a question of Krattenthaler, who noted that a bijection defined by the unmodified Fomin algorithm obviously commutes with inverses, and asked what the connection is between this bijection and $\phi^*$. Dans leur article sur l'équivalence de Wilf pour les classes de singletons, Backelin, West et Xin ont introduit une transformation $\phi^*$, définie par un processus itératif et opérant sur (tous) les placements complets de tours sur un plateau de Ferrers. Bousquet-Melou et Steingrimsson ont démontré l'analogue du résultat principal de Backelin, West et Xin dans le contexte d'involutions, et pour ce faire ont eu besoin de démontrer que $\phi^*$ commute avec l'opération inverse. La preuve de cette commutativité est longue et difficile, et Bousquet-Melou et Steingrômsson se demandèrent s'il n'était pas possible de reformuler $\phi^*$ de sorte que le resultat devienne évident. Dans le présent article, nous proposons une telle reformulation de $\phi^*$ en modifiant l'algorithme de croissance de diagramme de Fomin. Cette reformulation répond également à une question de Krattenthaler, qui, remarquant qu'une bijection définie par l'algorithme de Fomin non modifié commute évidemment avec l'opération inverse, se demanda quel était le rapport entre cette bijection et $\phi^*$.

scholarly journals Rationality, Irrationality, and Wilf Equivalence in Generalized Factor Order

10.37236/88 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Sergey Kitaev ◽  
Jeffrey Liese ◽  
Jeffrey Remmel ◽  
Bruce E. Sagan
Keyword(s):  
Generating Function ◽  
Ordered Set ◽  
Free Monoid ◽  
Recursive Formula ◽  
Finite State Automaton ◽  
Finite State ◽  
Pumping Lemma ◽  
Wilf Equivalence ◽  
Positive Integers ◽  
Factor Order

Let $P$ be a partially ordered set and consider the free monoid $P^*$ of all words over $P$. If $w,w'\in P^*$ then $w'$ is a factor of $w$ if there are words $u,v$ with $w=uw'v$. Define generalized factor order on $P^*$ by letting $u\le w$ if there is a factor $w'$ of $w$ having the same length as $u$ such that $u\le w'$, where the comparison of $u$ and $w'$ is done componentwise using the partial order in $P$. One obtains ordinary factor order by insisting that $u=w'$ or, equivalently, by taking $P$ to be an antichain. Given $u\in P^*$, we prove that the language ${\cal F}(u)=\{w\ :\ w\ge u\}$ is accepted by a finite state automaton. If $P$ is finite then it follows that the generating function $F(u)=\sum_{w\ge u} w$ is rational. This is an analogue of a theorem of Björner and Sagan for generalized subword order. We also consider $P={\Bbb P}$, the positive integers with the usual total order, so that $P^*$ is the set of compositions. In this case one obtains a weight generating function $F(u;t,x)$ by substituting $tx^n$ each time $n\in{\Bbb P}$ appears in $F(u)$. We show that this generating function is also rational by using the transfer-matrix method. Words $u,v$ are said to be Wilf equivalent if $F(u;t,x)=F(v;t,x)$ and we prove various Wilf equivalences combinatorially. Björner found a recursive formula for the Möbius function of ordinary factor order on $P^*$. It follows that one always has $\mu(u,w)=0,\pm1$. Using the Pumping Lemma we show that the generating function $M(u)=\sum_{w\ge u} |\mu(u,w)| w$ can be irrational.

scholarly journals Pattern avoidance in partial permutations (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Anders Claesson ◽  
Vít Jelínek ◽  
Eva Jelínková ◽  
Sergey Kitaev
Keyword(s):  
Equivalence Class ◽  
Arbitrary Number ◽  
Pattern Avoidance ◽  
Permutation Patterns ◽  
Nous Montrons ◽  
International Audience ◽  
Wilf Equivalence ◽  
Partial Words

International audience Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A $\textit{partial permutation of length n with k holes}$ is a sequence of symbols $\pi = \pi_1 \pi_2 \cdots \pi_n$ in which each of the symbols from the set $\{1,2,\ldots,n-k\}$ appears exactly once, while the remaining $k$ symbols of $\pi$ are "holes''. We introduce pattern-avoidance in partial permutations and prove that most of the previous results on Wilf equivalence of permutation patterns can be extended to partial permutations with an arbitrary number of holes. We also show that Baxter permutations of a given length $k$ correspond to a Wilf-type equivalence class with respect to partial permutations with $(k-2)$ holes. Lastly, we enumerate the partial permutations of length $n$ with $k$ holes avoiding a given pattern of length at most four, for each $n \geq k \geq 1$. Nous introduisons un concept de permutations partielles. $\textit{Une permutation partielle de longueur n avec k trous}$ est une suite finie de symboles $\pi = \pi_1 \pi_2 \cdots \pi_n$ dans laquelle chaque nombre de l'ensemble $\{1,2,\ldots,n-k\}$ apparaît précisément une fois, tandis que les $k$ autres symboles de $\pi$ sont des "trous''. Nous introduisons l'étude des permutations partielles à motifs exclus et nous montrons que la plupart des résultats sur l'équivalence de Wilf peuvent être généralisés aux permutations partielles avec un nombre arbitraire de trous. De plus, nous montrons que les permutations de Baxter d'une longueur donnée $k$ forment une classe d'équivalence du type Wilf par rapport aux permutations partielles avec $(k-2)$ trous. Enfin, nous présentons l'énumération des permutations partielles de longueur $n$ avec $k$ trous qui évitent un motif de longueur $\ell \leq 4$, pour chaque $n \geq k \geq 1$.

scholarly journals Patterns in matchings and rook placements

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Jonathan Bloom ◽  
Sergi Elizalde
Keyword(s):  
Generating Functions ◽  
Direct Proof ◽  
Pattern Avoidance ◽  
Set Partitions ◽  
Rook Placements ◽  
International Audience ◽  
Wilf Equivalence ◽  
Diagram Representation ◽  
Ferrers Boards

International audience Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards. We enumerate 312-avoiding matchings and partitions, obtaining algebraic generating functions, unlike in the 321-avoiding (i.e., 3-noncrossing) case. Our approach also provides a more direct proof of a formula of Bóna for the number of 1342-avoiding permutations. Additionally, we give a bijection proving the shape-Wilf-equivalence of the patterns 321 and 213 which simplifies existing proofs by Backelin–West–Xin and Jelínek.

scholarly journals Pattern-Avoidance in Binary Fillings of Grid Shapes (short version)

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Alexey Spiridonov
Keyword(s):  
Young Diagram ◽  
Bipartite Graphs ◽  
Pattern Avoidance ◽  
Short Version ◽  
Square Grid ◽  
Acyclic Orientations ◽  
Permutation Pattern ◽  
Nous Montrons ◽  
International Audience ◽  
Wilf Equivalence

International audience A $\textit{grid shape}$ is a set of boxes chosen from a square grid; any Young diagram is an example. This paper considers a notion of pattern-avoidance for $0-1$ fillings of grid shapes, which generalizes permutation pattern-avoidance. A filling avoids some patterns if none of its sub-shapes equal any of the patterns. We focus on patterns that are $\textit{pairs}$ of $2 \times 2$ fillings. For some shapes, fillings that avoid specific $2 \times 2$ pairs are in bijection with totally nonnegative Grassmann cells, or with acyclic orientations of bipartite graphs. We prove a number of results analogous to Wilf-equivalence for these objects ―- that is, we show that for certain classes of shapes, some pattern-avoiding fillings are equinumerous with others. Une $\textit{forme de grille}$ est un ensemble de cases choisies dans une grille carrée; un diagramme de Young en est un exemple. Cet article considère une notion de motif exclu pour un remplissage d'une forme de grille par des $0$ et des $1$, qui généralise la notion correspondante pour les permutations. Un remplissage évite certains motifs si aucune de ses sous-formes n'est égale à un motif. Nous nous concentrons sur les motifs qui sont des $\textit{paires de remplissages}$ de taille $2 \times 2$. Pour certaines formes, les remplissages évitant certaines paires de taille $2 \times 2$ sont en bijection avec les cellules de Grassmann totalement positives, ou bien avec les orientations acycliques de graphes bipartis. Nous démontrons plusieurs résultats analogues à l'équivalence de Wilf pour ces objets ―- c'est-à-dire, nous montrons que, pour certaines classes de formes, des remplissages évitant un motif donné sont en nombre égal à d'autres remplissages.

Somethin’ about Scandinavia: Danish Shorts on the Post-war International Scene

2018 ◽  
Author(s):  
C. Claire Thomson
Keyword(s):  
United Nations ◽  
International Collaboration ◽  
Marshall Plan ◽  
The Third ◽  
Animated Film ◽  
International Audience ◽  
Post War ◽  
Nordic Cooperation

Building on the picture of post-war Anglo-Danish documentary collaboration established in the previous chapter, this chapter examines three cases of international collaboration in which Dansk Kulturfilm and Ministeriernes Filmudvalg were involved in the late 1940s and 1950s. They Guide You Across (Ingolf Boisen, 1949) was commissioned to showcase Scandinavian cooperation in the realm of aviation (SAS) and was adopted by the newly-established United Nations Film Board. The complexities of this film’s production, funding and distribution are illustrative of the activities of the UN Film Board in its first years of operation. The second case study considers Alle mine Skibe (All My Ships, Theodor Christensen, 1951) as an example of a film commissioned and funded under the auspices of the Marshall Plan. This US initiative sponsored informational films across Europe, emphasising national solutions to post-war reconstruction. The third case study, Bent Barfod’s animated film Noget om Norden (Somethin’ about Scandinavia, 1956) explains Nordic cooperation for an international audience, but ironically exposed some gaps in inter-Nordic collaboration in the realm of film.

Conclusion

2018 ◽  
Author(s):  
Alistair Fox
Keyword(s):  
National Identity ◽  
New Zealand ◽  
Identity Formation ◽  
Essential Role ◽  
Coming Of Age ◽  
Financial Returns ◽  
Self Recognition ◽  
International Audience ◽  
National Identity Formation

The conclusion reaffirms the essential role played by cinema generally, and the coming-of-age genre in particular, in the process of national identity formation, because of its effectiveness in facilitating self-recognition and self-experience through a process of triangulation made possible, for the most part, by a dialogue with some of the nation’s most iconic works of literature. This section concludes by point out the danger posed, however, by an observable trend toward generic standardization in New Zealand films motivated by a desire to appeal to an international audience out of consideration for the financial returns expected by funding bodies under current regimes.

Early Cinema in Scotland

2018 ◽  
Keyword(s):  
Social History ◽  
Small Towns ◽  
Early Cinema ◽  
Popular Song ◽  
Feature Film ◽  
Local Newspapers ◽  
The Social ◽  
Production History ◽  
History Of ◽  
International Audience

This collection of essays, drawn from a three-year AHRC research project, provides a detailed context for the history of early cinema in Scotland from its inception in 1896 till the arrival of sound in the early 1930s. It details the movement from travelling fairground shows to the establishment of permanent cinemas, and from variety and live entertainment to the dominance of the feature film. It addresses the promotion of cinema as a socially ‘useful’ entertainment, and, distinctively, it considers the early development of cinema in small towns as well as in larger cities. Using local newspapers and other archive sources, it details the evolution and the diversity of the social experience of cinema, both for picture goers and for cinema staff. In production, it examines the early attempts to establish a feature film production sector, with a detailed production history of Rob Roy (United Films, 1911), and it records the importance, both for exhibition and for social history, of ‘local topicals’. It considers the popularity of Scotland as an imaginary location for European and American films, drawing their popularity from the international audience for writers such as Walter Scott and J.M. Barrie and the ubiquity of Scottish popular song. The book concludes with a consideration of the arrival of sound in Scittish cinemas. As an afterpiece, it offers an annotated filmography of Scottish-themed feature films from 1896 to 1927, drawing evidence from synopses and reviews in contemporary trade journals.

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