اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدTri-Partitions and Bases of an Ordered Complex
86
0
0.0
(
0
)
نشر من قبل
Herbert Edelsbrunner
تاريخ النشر
2021
مجال البحث
الهندسة المعلوماتية
والبحث باللغة
English
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholz-Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, $K$, and every dimension, $p$, there is a partition of the set of $p$-cells into a maximal $p$-tree, a maximal $p$-cotree, and a collection of $p$-cells whose cardinality is the $p$-th Betti number of $K$. Given an ordering of the $p$-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.
قيم البحث
اقرأ أيضاً
The symmetric group $mathfrak{S}_n$ acts on the polynomial ring $mathbb{Q}[mathbf{x}_n] = mathbb{Q}[x_1, dots, x_n]$ by variable permutation. The invariant ideal $I_n$ is the ideal generated by all $mathfrak{S}_n$-invariant polynomials with vanishing
constant term. The quotient $R_n = frac{mathbb{Q}[mathbf{x}_n]}{I_n}$ is called the coinvariant algebra. The coinvariant algebra $R_n$ has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization $I_{n,k} subseteq mathbb{Q}[mathbf{x}_n]$ of the ideal $I_n$ indexed by two positive integers $k leq n$. The corresponding quotient $R_{n,k} := frac{mathbb{Q}[mathbf{x}_n]}{I_{n,k}}$ carries a graded action of $mathfrak{S}_n$ and specializes to $R_n$ when $k = n$. We generalize many of the nice properties of $R_n$ to $R_{n,k}$. In particular, we describe the Hilbert series of $R_{n,k}$, give extensions of the Artin and Garsia-Stanton monomial bases of $R_n$ to $R_{n,k}$, determine the reduced Grobner basis for $I_{n,k}$ with respect to the lexicographic monomial order, and describe the graded Frobenius series of $R_{n,k}$. Just as the combinatorics of $R_n$ are controlled by permutations in $mathfrak{S}_n$, we will show that the combinatorics of $R_{n,k}$ are controlled by ordered set partitions of ${1, 2, dots, n}$ with $k$ blocks. The {em Delta Conjecture} of Haglund, Remmel, and Wilson is a generalization of the Shuffle Conjecture in the theory of diagonal coinvariants. We will show that the graded Frobenius series of $R_{n,k}$ is (up to a minor twist) the $t = 0$ specialization of the combinatorial side of the Delta Conjecture. It remains an open problem to give a bigraded $mathfrak{S}_n$-module $V_{n,k}$ whose Frobenius image is even conjecturally equal to any of the expressions in the Delta Conjecture; our module $R_{n,k}$ solves this problem in the specialization $t = 0$.
We study statistics on ordered set partitions whose generating functions are related to $p,q$-Stirling numbers of the second kind. The main purpose of this paper is to provide bijective proofs of all the conjectures of stein (Arxiv:math.CO/0605670).
Our basic idea is to encode ordered partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. We also give a partition version of MacMahons theorem on the equidistribution of the statistics inversion number and major index on words.
For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in ter
ms of the (reduced) type of $k$-divisible noncrossing partitions. Our work extends Haimans notion of a parking function symmetric function.
In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed parti
tions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of third order $f(q)$, $phi(q)$ and $psi(q)$. An identity of Ramanujan is proved combinatorially. Several new identities are also established.
We study two families of probability measures on integer partitions, which are Schur measures with parameters tuned in such a way that the edge fluctuations are characterized by a critical exponent different from the generic $1/3$. We find that the f
irst part asymptotically follows a higher-order analogue of the Tracy-Widom GUE distribution, previously encountered by Le Doussal, Majumdar and Schehr in quantum statistical physics. We also compute limit shapes, and discuss an exact mapping between one of our families and the multicritical unitary matrix models introduced by Periwal and Shevitz.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
