scholarly journals Reduction formulae from the factorization Theorem of Littlewood-Richardson polynomials by King, Tollu and Toumazet

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Soojin Cho ◽  
Eun-Kyoung Jung ◽  
Dongho Moon
Keyword(s):  
Factorization Theorem ◽  
Reduction Formula ◽  
International Audience ◽  
Classical Reduction ◽  
Special Case

International audience The factorization theorem by King, Tollu and Toumazet gives four different reduction formulae of Littlewood-Richardson coefficients. One of them is the classical reduction formula of the first type while others are new. Moreover, the classical reduction formula of the second type is not a special case of KTT theorem. We give combinatorial proofs of reduction formulae in terms of tableaux or hives. The proofs for the cases $r=1, 2, n-2$ in terms of tableaux and the proof for the classical reduction formula of the second type in terms of hives are new. Le Théorème de factorisation par King, Tollu, et Toumazet donne quatre formules différentes de réduction des coefficients de Littlewood-Richardson. L'une d'entre eux est la formule classique de réduction du premier type alors que les trois autres sont nouvelles. De plus, la formule classique de réduction du deuxième type n'est pas un cas spécial du KTT théorème. Nous donnons preuves combinatoires des formules de réduction en termes de tableaux ou $\textit{hives}$. Les preuves pour les cas $r = 1, 2, n-2$ en termes de tableaux et la preuve pour la formule classique de réduction du deuxième type en termes de $\textit{hives}$ sont nouvelles.

scholarly journals Constrained exchangeable partitions

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Alexander Gnedin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Partitions ◽  
Type Representation ◽  
International Audience ◽  
De Finetti ◽  
Infinite Set ◽  
Special Case

International audience For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.

scholarly journals On the monotone hook hafnian conjecture

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Mirkó Visontai
Keyword(s):  
Graph Polynomials ◽  
Real Roots ◽  
Threshold Graphs ◽  
Multivariate Generalization ◽  
International Audience ◽  
Special Case

International audience We investigate a conjecture of Haglund that asserts that certain graph polynomials have only real roots. We prove a multivariate generalization of this conjecture for the special case of threshold graphs. Nous étudions une conjecture de Haglund qui affirme que certaines polynômes des graphes ont uniquement des racines réelles. Nous prouvons une généralisation multivariée de cette conjecture pour le cas particulier des graphes à seuil.

scholarly journals Enumerating (2+2)-free posets by the number of minimal elements and other statistics

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel
Keyword(s):  
Generating Function ◽  
Minimal Elements ◽  
Nous Montrons ◽  
International Audience ◽  
Ascent Sequences ◽  
Special Case

International audience A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. In a recent paper, Bousquet-Mélou et al. found, using so called ascent sequences, the generating function for the number of (2+2)-free posets: $P(t)=∑_n≥ 0 ∏_i=1^n ( 1-(1-t)^i)$. We extend this result by finding the generating function for (2+2)-free posets when four statistics are taken into account, one of which is the number of minimal elements in a poset. We also show that in a special case when only minimal elements are of interest, our rather involved generating function can be rewritten in the form $P(t,z)=∑_n,k ≥0 p_n,k t^n z^k = 1+ ∑_n ≥0\frac{zt}{(1-zt)^n+1}∏_i=1^n (1-(1-t)^i)$ where $p_n,k$ equals the number of (2+2)-free posets of size $n$ with $k$ minimal elements. Un poset sera dit (2+2)-libre s'il ne contient aucun sous-poset isomorphe à 2+2, l'union disjointe de deux chaînes à deux éléments. Dans un article récent, Bousquet-Mélou et al. ont trouvé, à l'aide de "suites de montées'', la fonction génératrice des nombres de posets (2+2)-libres: c'est $P(t)=∑_n≥ 0 ∏_i=1^n ( 1-(1-t)^i)$. Nous étendons ce résultat en trouvant la fonction génératrice des posets (\textrm2+2)-libres rendant compte de quatre statistiques, dont le nombre d'éléments minimaux du poset. Nous montrons aussi que lorsqu'on ne s'intéresse qu'au nombre d'éléments minimaux, notre fonction génératrice assez compliquée peut être simplifiée en$P(t,z)=∑_n,k ≥0 p_n,k t^n z^k = 1+ ∑_n ≥0\frac{zt}{(1-zt)^n+1}∏_i=1^n (1-(1-t)^i)$, où $p_n,k$ est le nombre de posets (2+2)-libres de taille $n$ avec $k$ éléments minimaux.

scholarly journals On the minimal distance of a polynomial code

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Peter Pal Pach ◽  
Csaba Szabo
Keyword(s):  
Finite Subset ◽  
60Th Birthday ◽  
Theoretical Problem ◽  
Minimal Distance ◽  
Distribution Of Primes ◽  
Special Issue ◽  
Theoretical Methods ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Special Case

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For a polynomial f(x) is an element of Z(2)[x] it is natural to consider the near-ring code generated by the polynomials f circle x, f circle x(2) ,..., f circle x(k) as a vectorspace. It is a 19 year old conjecture of Gunter Pilz that for the polynomial f (x) - x(n) broken vertical bar x(n-1) broken vertical bar ... broken vertical bar x the minimal distance of this code is n. The conjecture is equivalent to the following purely number theoretical problem. Let (m) under bar = \1, 2 ,..., m\ and A subset of N be an arbitrary finite subset of N. Show that the number of products that occur odd many times in (n) under bar. A is at least n. Pilz also formulated the conjecture for the special case when A = (k) under bar. We show that for A = (k) under bar the conjecture holds and that the minimal distance of the code is at least n/(log n)(0.223). While proving the case A = (k) under bar we use different number theoretical methods depending on the size of k (respect to n). Furthermore, we apply several estimates on the distribution of primes.

scholarly journals On the class number formula of certain real quadratic fields

2013 ◽  
Vol Volume 36 ◽  
Author(s):  
K Chakraborty ◽  
S Kanemitsu ◽  
T Kuzumaki
Keyword(s):  
Class Number ◽  
Character Sums ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Classical Class ◽  
Class Number Formula ◽  
Number Formula ◽  
International Audience ◽  
Special Case ◽  
Real Quadratic

International audience In this note we give an alternate expression of class number formula for real quadratic fields with discriminant $d \equiv 5\, {\rm mod}\, 8$. %Dirichlet's classical class number formula for real quadratic fields expresses `class number' in somewhat `transcend' manner, which was simplified by P. Chowla in the special case when the discriminant $d = p \equiv 5\,{\rm mod}\, 8$ is a prime. We use another form of class number formula and transform it using Dirichlet's $1/4$-th character sums. Our result elucidates other generalizations of the class number formula by Mitsuhiro, Nakahara and Uhera for general real quadratic fields.

scholarly journals Dual combinatorics of zonal polynomials

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Valentin Féray ◽  
Piotr Sniady
Keyword(s):  
Irreducible Character ◽  
Symmetric Functions ◽  
Symmetric Groups ◽  
Combinatorial Formula ◽  
Zonal Polynomials ◽  
Power Sums ◽  
Power Sum ◽  
Recent Developments ◽  
International Audience ◽  
Special Case

International audience In this paper we establish a new combinatorial formula for zonal polynomials in terms of power-sums. The proof relies on the sign-reversing involution principle. We deduce from it formulas for zonal characters, which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. These formulas are analogs of recent developments on irreducible character values of symmetric groups. The existence of such formulas could have been predicted from the work of M. Lassalle who formulated two positivity conjectures for Jack characters, which we prove in the special case of zonal polynomials. Dans cet article, nous établissons une nouvelle formule combinatoire pour les polynômes zonaux en fonction des fonctions puissance. La preuve utilise le principe de l'involution changeant les signes. Nous en déduisons des formules pour les caractères zonaux, qui sont définis comme les coefficients des polynômes zonaux écrits sur la base des fonctions puissance, normalisés de manière appropriée. Ces formules sont des analogues de développements récents sur les caractères du groupe symétrique. L'existence de telles formules aurait pu être prédite à partir des travaux de M. Lassalle, qui a proposé deux conjectures de positivité sur les caractères de Jack, que nous prouvons dans le cas particulier des polynômes zonaux.

scholarly journals Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Coxeter Groups ◽  
Reflection Group ◽  
Sperner Property ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Noncrossing Partition ◽  
International Audience ◽  
Partition Lattices ◽  
Special Case

International audience We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.

scholarly journals Are even maps on surfaces likely to be bipartite?

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy
Keyword(s):  
Asymptotic Behaviour ◽  
Planar Map ◽  
Generating Series ◽  
Finite Set ◽  
Positive Genus ◽  
International Audience ◽  
Special Case

International audience It is well known that a planar map is bipartite if and only if all its faces have even degree (what we call an even map). In this paper, we show that rooted even maps of positive genus $g$ chosen uniformly at random are bipartite with probability tending to $4^{−g}$ when their size goes to infinity. Loosely speaking, we show that each of the $2g$ fundamental cycles of the surface of genus $g$ contributes a factor $\frac{1}{2}$ to this probability.We actually do more than that: we obtain the explicit asymptotic behaviour of the number of even maps and bipartite maps of given genus with any finite set of allowed face degrees. This uses a generalisation of the Bouttier-Di Francesco-Guitter bijection to the case of positive genus, a decomposition inspired by previous works of Marcus, Schaeffer and the author, and some involved manipulations of generating series counting paths. A special case of our results implies former conjectures of Gao.

scholarly journals On the link pattern distribution of quarter-turn symmetric FPL configurations

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Philippe Duchon
Keyword(s):  
Nous Obtenons ◽  
Plane Partitions ◽  
New Class ◽  
International Audience ◽  
Pattern Distribution ◽  
Special Case ◽  
Link Pattern

International audience We present new conjectures on the distribution of link patterns for fully-packed loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation, extending previous conjectures of Razumov and Stroganov and of de Gier. We prove a special case, showing that the link pattern that is conjectured to be the rarest does have the prescribed probability. As a byproduct, we get a formula for the enumeration of a new class of quasi-symmetry of plane partitions. Nous présentons de nouvelles conjectures portant sur la distribution des schémas de couplage des configurations de boucles compactes (FPL) invariantes, ou presque invariantes, par une rotation d'un quart de tour. Ces nouvelles conjectures étendent des conjectures précédentes dues à Razumov et Stroganov et à de Gier. Dans chaque cas, nous prouvons un cas particulier, en démontrant que le schéma de couplage conjecturé pour être le plus rare a effectivement la probabilité prédite. Nous obtenons également une formule pour l'énumération d'une nouvelle classe de quasi-symétrie de partitions planes.

scholarly journals Limit laws for a class of diminishing urn models.

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Random Variables ◽  
Urn Models ◽  
Limit Laws ◽  
Sampling Without Replacement ◽  
International Audience ◽  
Special Instance ◽  
Special Case

International audience In this work we analyze a class of diminishing 2×2 Pólya-Eggenberger urn models with ball replacement matrix M given by $M= \binom{ -a \,0}{c -d}, a,d∈\mathbb{N}$ and $c∈\mathbb{N} _0$. We obtain limit laws for this class of 2×2 urns by giving estimates for the moments of the considered random variables. As a special instance we obtain limit laws for the pills problem, proposed by Knuth and McCarthy, which corresponds to the special case $a=c=d=1$. Furthermore, we also obtain limit laws for the well known sampling without replacement urn, $a=d=1$ and $c=0$, and corresponding generalizations, $a,d∈\mathbb{N}$ and $c=0$.

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