scholarly journals A Bijection for Directed-Convex Polyominoes

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Alberto Del Lungo ◽  
Massimo Mirolli ◽  
Renzo Pinzani ◽  
Simone Rinaldi
Keyword(s):  
Fixed Number ◽  
Lattice Paths ◽  
Bijective Proof ◽  
International Audience

International audience In this paper we consider two classes of lattice paths on the plane which use \textitnorth, \textiteast, \textitsouth,and \textitwest unitary steps, beginningand ending at (0,0).We enumerate them according to the number ofsteps by means of bijective arguments; in particular, we apply the cycle lemma.Then, using these results, we provide a bijective proof for the number of directed-convex polyominoes having a fixed number of rows and columns.

scholarly journals A direct bijective proof of the hook-length formula

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Jean-Christophe Novelli ◽  
Igor Pak ◽  
Alexander V. Stoyanovskii
Keyword(s):  
Young Tableaux ◽  
Hook Length ◽  
Bijective Proof ◽  
International Audience ◽  
Standard Young Tableaux

International audience This paper presents a new proof of the hook-length formula, which computes the number of standard Young tableaux of a given shape. After recalling the basic definitions, we present two inverse algorithms giving the desired bijection. The next part of the paper presents the proof of the bijectivity of our construction. The paper concludes with some examples.

scholarly journals Random sampling of lattice paths with constraints, via transportation

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Lucas Gerin
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Random Sampling ◽  
Optimal Transport ◽  
Lattice Paths ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Contraction Property ◽  
International Audience

International audience We build and analyze in this paper Markov chains for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. These chains are easy to implement, and sample an "almost" uniform path of length $n$ in $n^{3+\epsilon}$ steps. This bound makes use of a certain $\textit{contraction property}$ of the Markov chain, and is proved with an approach inspired by optimal transport.

scholarly journals A bijective proof of Macdonald's reduced word formula

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Sara Billey ◽  
Alexander Holroyd ◽  
Benjamin Young
Keyword(s):  
Reduced Word ◽  
Bijective Proof ◽  
International Audience ◽  
Principal Specialization

International audience We describe a bijective proof of Macdonald's reduced word identity using pipe dreams and Little's bumping algorithm. The proof extends to a principal specialization of the identity due to Fomin and Stanley. Our bijective tools also allow us to address a problem posed by Fomin and Kirillov from 1997, using work of Wachs, Lenart and Serrano- Stump.

scholarly journals Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's result

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Valentin Féray ◽  
Ekaterina A. Vassilieva
Keyword(s):  
Stirling Numbers ◽  
Algebraic Methods ◽  
Combinatorial Proof ◽  
Nous Obtenons ◽  
Long Cycle ◽  
Bijective Proof ◽  
International Audience

International audience We look at the number of permutations $\beta$ of $[N]$ with $m$ cycles such that $(1 2 \ldots N) \beta^{-1}$ is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size $N+1$. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account. Nous étudions le nombre de permutations $\beta$ de $[N]$ avec $m$ cycles telles que $(1 2 \ldots N) \beta^{-1}$ a un seul cycle. Ces nombres apparaissent en tant que coefficients des monômes linéaires des polynômes de Kerov et de Stanley. À l'aide de méthodes algébriques, D. Zagier a trouvé une connexion inattendue avec les nombres de Stirling de taille $N+1$. Nous présentons ici la première preuve combinatoire de son résultat, en introduisant une nouvelle bijection entre des cartes partitionnées et des arbres épineux. De plus, nous obtenons un résultat plus fin, prenant en compte le type des permutations.

scholarly journals Top Coefficients of the Denumerant

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Velleda Baldoni ◽  
Nicole Berline ◽  
Brandon Dutra ◽  
Matthias Köppe ◽  
Michele Vergne ◽  
...  
Keyword(s):  
Nonnegative Integer ◽  
Fixed Number ◽  
Lattice Points ◽  
Combinatorial Number Theory ◽  
International Audience ◽  
Integer Solutions ◽  
Combinatorial Function ◽  
Rational Generating Function ◽  
Rational Generating Functions ◽  
The Right

International audience For a given sequence $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\alpha)(t)$ that counts the nonnegative integer solutions of the equation $\alpha_1x_1+\alpha_2 x_2+ \ldots+ \alpha_Nx_N+ \alpha_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying nonnegative integer. It is well-known that $E(\alpha)(t)$ is a quasipolynomial function of $t$ of degree $N$. In combinatorial number theory this function is known as the $\textit{denumerant}$. Our main result is a new algorithm that, for every fixed number $k$, computes in polynomial time the highest $k+1$ coefficients of the quasi-polynomial $E(\alpha)(t)$ as step polynomials of $t$. Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for $E(\alpha)(t)$ and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Experiments using a $\texttt{MAPLE}$ implementation will be posted separately. Considérons une liste $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ de $N+1$ entiers positifs. Le dénumérant $E(\alpha)(t)$ est lafonction qui compte le nombre de solutions en entiers positifs ou nuls de l’équation $\sum^{N+1}_{i=1}x_i\alpha_i=t$, où $t$ varie dans les entiers positifs ou nuls. Il est bien connu que cette fonction est une fonction quasi-polynomiale de $t$, de degré $N$. Nous donnons un nouvel algorithme qui calcule, pour chaque entier fixé $k$ (mais $N$ n’est pas fixé, les $k+1$ plus hauts coefficients du quasi-polynôme $E(\alpha)(t)$ en termes de fonctions en dents de scie. Notre algorithme utilise la structure d’ensemble partiellement ordonné des pôles de la fonction génératrice de $E(\alpha)(t)$. Les $k+1$ plus hauts coefficients se calculent à l’aide de fonctions génératrices de points entiers dans des cônes polyèdraux de dimension inférieure ou égale à $k$.

scholarly journals An extension to overpartitions of Rogers-Ramanujan identities for even moduli

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Sylvie Corteel ◽  
Jeremy Lovejoy ◽  
Olivier Mallet
Keyword(s):  
Generating Functions ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Lattice Paths ◽  
Infinite Products ◽  
International Audience

International audience We investigate class of well-poised basic hypergeometric series $\tilde{J}_{k,i}(a;x;q)$, interpreting these series as generating functions for overpartitions defined by multiplicity conditions. We also show how to interpret the $\tilde{J}_{k,i}(a;1;q)$ as generating functions for overpartitions whose successive ranks are bounded, for overpartitions that are invariant under a certain class of conjugations, and for special restricted lattice paths. We highlight the cases $(a,q) \to (1/q,q)$, $(1/q,q^2)$, and $(0,q)$, where some of the functions $\tilde{J}_{k,i}(a;x;q)$ become infinite products. The latter case corresponds to Bressoud's family of Rogers-Ramanujan identities for even moduli.

scholarly journals The Rees product of the cubical lattice with the chain

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Patricia Muldoon ◽  
Margaret A. Readdy
Keyword(s):  
Möbius Function ◽  
Explicit Formulas ◽  
Bijective Proof ◽  
Mobius Function ◽  
International Audience

International audience We study enumerative and homological properties of the Rees product of the cubical lattice with the chain. We give several explicit formulas for the Möbius function. The last formula is expressed in terms of the permanent of a matrix and is given by a bijective proof. Nous étudions des propriétés énumératives et homologiques du produit de Rees du treillis cubique avec la chaîne. Nous donnons plusieurs formules explicites de la fonction de Möbius de ce poset. La dernière de ces formules est exprimée en termes du permanent d’une matrice et le résultat est donné par une preuve bijective.

scholarly journals Osculating Paths and Oscillating Tableaux

10.37236/731 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Roger E. Behrend
Keyword(s):  
Young Diagram ◽  
Fixed Number ◽  
Lattice Points ◽  
Lattice Paths ◽  
Square Lattice ◽  
Vertex Model ◽  
Young Diagrams ◽  
Primary Interest ◽  
Alternating Sign Matrices ◽  
Positive Integers

The combinatorics of certain tuples of osculating lattice paths is studied, and a relationship with oscillating tableaux is obtained. The paths being considered have fixed start and end points on respectively the lower and right boundaries of a rectangle in the square lattice, each path can take only unit steps rightwards or upwards, and two different paths within a tuple are permitted to share lattice points, but not to cross or share lattice edges. Such path tuples correspond to configurations of the six-vertex model of statistical mechanics with appropriate boundary conditions, and they include cases which correspond to alternating sign matrices. Of primary interest here are path tuples with a fixed number $l$ of vacancies and osculations, where vacancies or osculations are points of the rectangle through which respectively no or two paths pass. It is shown that there exist natural bijections which map each such path tuple $P$ to a pair $(t,\eta)$, where $\eta$ is an oscillating tableau of length $l$ (i.e., a sequence of $l+1$ partitions, starting with the empty partition, in which the Young diagrams of successive partitions differ by a single square), and $t$ is a certain, compatible sequence of $l$ weakly increasing positive integers. Furthermore, each vacancy or osculation of $P$ corresponds to a partition in $\eta$ whose Young diagram is obtained from that of its predecessor by respectively the addition or deletion of a square. These bijections lead to enumeration formulae for tuples of osculating paths involving sums over oscillating tableaux.

scholarly journals The Size of One-Way Cellular Automata

2010 ◽  
Vol DMTCS Proceedings vol. AL,... (Proceedings) ◽  
Author(s):  
Martin Kutrib ◽  
Jonas Lefèvre ◽  
Andreas Malcher
Keyword(s):  
Cellular Automata ◽  
Real Time ◽  
Finite Automata ◽  
Fixed Number ◽  
Point Of View ◽  
Upper And Lower Bounds ◽  
Worst Case ◽  
Order Of Magnitude ◽  
International Audience ◽  
Number Of Cells

International audience We investigate the descriptional complexity of basic operations on real-time one-way cellular automata with an unbounded as well well as a fixed number of cells. The size of the automata is measured by their number of states. Most of the bounds shown are tight in the order of magnitude, that is, the sizes resulting from the effective constructions given are optimal with respect to worst case complexity. Conversely, these bounds also show the maximal savings of size that can be achieved when a given minimal real-time OCA is decomposed into smaller ones with respect to a given operation. From this point of view the natural problem of whether a decomposition can algorithmically be solved is studied. It turns out that all decomposition problems considered are algorithmically unsolvable. Therefore, a very restricted cellular model is studied in the second part of the paper, namely, real-time one-way cellular automata with a fixed number of cells. These devices are known to capture the regular languages and, thus, all the problems being undecidable for general one-way cellular automata become decidable. It is shown that these decision problems are $\textsf{NLOGSPACE}$-complete and thus share the attractive computational complexity of deterministic finite automata. Furthermore, the state complexity of basic operations for these devices is studied and upper and lower bounds are given.

scholarly journals Bijections for lattice paths between two boundaries

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Sergi Elizalde ◽  
Martin Rubey
Keyword(s):  
Joint Distribution ◽  
Tutte Polynomial ◽  
Lattice Paths ◽  
Dyck Paths ◽  
International Audience ◽  
Pattern Avoiding Permutations

International audience We prove that on the set of lattice paths with steps $N=(0,1)$ and $E=(1,0)$ that lie between two boundaries $B$ and $T$, the two statistics `number of $E$ steps shared with $B$' and `number of $E$ steps shared with $T$' have a symmetric joint distribution. We give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps $S=(0,-1)$ at prescribed $x$-coordinates. We also show that a similar equidistribution result for other path statistics follows from the fact that the Tutte polynomial of a matroid is independent of the order of its ground set. Finally, we extend the two theorems to $k$-tuples of paths between two boundaries, and we give some applications to Dyck paths, generalizing a result of Deutsch, and to pattern-avoiding permutations. On montre que, sur l'ensemble des chemins avec des pas $N=(0,1)$ et $E=(1,0)$ qui se trouvent entre deux chemins donnés $B$ et $T$, les deux statistiques"`nombre des pas $E$ en commun avec $B$" et "nombre des pas $E$ en commun avec $T$" ont une distribution conjointe symétrique. On donne une involution qui échange ces deux statistiques, préserve quelques autres paramètres additionnels, et admet une généralisation à des chemins avec des pas $S=(0, -1)$ dans des positions données. On montre aussi un autre résultat d'équidistribution similaire, lié au polynôme de Tutte d'un matroïde. Finalement, on étend les deux théorèmes à $k$-tuples de chemins entre deux frontières, et on donne quelques applications aux chemins de Dyck, en généralisant un résultat de Deutsch, et aux permutations avec des motifs exclus.

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