What i mean by combinatorial geometry consists of problems in which one starts with a geometric figure say a polytope but then considers abstract incidence properties of it rather than its metric properties. This formula has been discovered by frame, robinson and thrall 2 in 1954, but only recently has a similar formula been proven in the more general case of. Pak, hook length formula and geometric combinatorics, sim. The latter is established by an explicit continuous volumepreserving piecewise linear map. His work on the foundations of combinatorics focused on the algebraic structures that lie behind diverse combinatorial areas, and created a new area of algebraic combinatorics. Hook length property of dcomplete posets via qintegrals the hook length formula for dcomplete posets states that the ppartition generating function for them is given by a product in terms of hook lengths. In this note, we give a simple and direct proof for the hook length formula. Hook length formula enumerative combinatorics of standard young tableaux hlf.
These are not the graphs of analytic geometry, but what are often described. Topics include enumerative, algebraic and geometric combinatorics on lattice polytopes, topological combinatorics, commutative algebra and toric varieties. An antichain is a subset of por, again, pitself in which no two of its elements are comparable. The surprising mathematics of longest increasing subsequences. Find materials for this course in the pages linked along the left. A related formula counts the number of semistandard young tableaux, which is a. Then the number of di erent permutations of all n objects is n. We are also grateful to birs, in banff, canada, for hosting the first two authors at the asymptotic algebraic combinatorics workshop in march 2019, where this paper was finalized. As the name suggests, however, it is broader than this. This section provides information on the readings for the topics covered in this course. A general formula to determine the number of ways an m nboard can be tiled with dominoes is known. Readers will find that this volume showcases current trends on lattice polytopes and stimulates further developments of many research areas surrounding this field. Anyone who is interested in modern analytic combinatorics will want to study this book.
Q s2 h s 1 since this was rst proved by frame, robinson and thrall, many di erent proofs have. The first author is thankful to alejandro morales and greta panova for numerous interesting conversations about the naruse hook length formula. The weighted hook length formula math user home pages. Most tiling and coloring problems fit into this class.
This volume contains the proceedings of the ams special session on discrete geometry and algebraic combinatorics held on january 11, 20, in san diego, california. Pakhook length formula and geometric combinatorics. This course is an introduction to algebraic combinatorics. Counting standard young tableaux hook length formula. Hook length formula how is hook length formula abbreviated. A similar thing happens with the column of permutations that start with \3. Chapter 1 elementary enumeration principles sequences theorem 1. Gill williamson chair alfano, joseph anthony, the module of diagonal harmonic polynomials 1994, adriano m. The proof requires a combination of combinatorial techniques, in particular a use of the hook length formula another important formula in combinatorics, in fact its currently the most highly voted answer to this math overflow question, and difficult analytic techniques complex analysis, hilbert transforms, the calculus of variations. The central result is the famous baikdeiftjohansson theorem that determines the asymptotic distribution of the length of the longest increasing subsequence of a random permutation, but many delicious topics are covered along the way. The naruse hook length formula is a recent general formula for the number of standard. The basic principle counting formulas the binomial theorem. Chapter 12 miscellaneous gems of algebraic combinatorics 231 12.
The symmetric group, its representations, and combinatorics. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The number f jsytjof standard young tableaux of shape has the celebrated hooklength formula hlf. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc. For this, we study the topics of permutations and combinations. The ones marked may be different from the article in the profile. Based on the ideas in ciocanfontanine, konvalinka and pak 2009, we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. The first proof is completely bijective, and in a special case gives a new short combinatorial proof of the hook length formula. We present here an elementary new proof which uses nothing more than the fundamental theorem of algebra. Eudml hook length formula and geometric combinatorics. Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. The formula was originally discovered by frame, robinson and thrall in frt based on earlier formula of thrall thr.
Hook length formula and geometric combinatorics citeseerx. Garsia chair allen, edward ernest, on a conjecture of procesi and a new basis of graded left regular representation 1991, adriano m. Combinatorics is often described brie y as being about counting, and indeed counting is a large part of combinatorics. We give a new proof of the hook length formula using qintegrals. We present an extensive survey of bijective proofs of classical partitions identities. The collection of articles in this volume is devoted to packings of metric spaces and related questions, and contains new results as well as surveys of some areas of discrete geometry.
Pdf hook length formula and geometric combinatorics. It is impossible to give a meaningful summary of the many facets of algebraic and geometric combinatorics in a writeup of this length. We give an algebraic and a combinatorial proof of naruses formula, by using factorial. The special case of an 8 8 board is already nontrivial. Permutation sign under the robinsonschensted correspondence. Lastly, we focus on the famous hook length formula giving the number of young tableaux of a certain shape. We present here a transparent proof of the hook length formula. The attached note gives a variant of a proof by bandlow 1 regarding hlf and has been streamlined so that it requires a mere two limits and shows a. In combinatorial mathematics, the hook length formula is a formula for the number of standard young tableaux whose shape is a given young diagram.
As such, the formula can be derived as a special case of the hook length formula. The hook length formula is a well known result expressing the number of standard tableaux of shape in terms of the lengths of the hooks in the diagram of. Various extensions and generalizations are added in the form of exercises. Due to its beauty and simplicity, the hook length formula has long attracted a great deal of attention from combinatorialists. Combinatorics counting an overview introductory example what to count lists permutations combinations. Algebraic and geometric methods in enumerative combinatorics. This formula has a number of combinatorial proofs, including purely bijective see fz, nps, pak, rem, zei. Giancarlo rota was one of the most original and colourful mathematicians of the 20th century. This cited by count includes citations to the following articles in scholar.
There are several geometric ways to think about this. The work of alfred young you01, you02 shows that f gives the dimension of the irreducible representation indexed by. The formula was discovered by frame, robinson and thrall in frt based on earlier results of young you, frobenius fro and thrall thr. The proof is done by a casebycase analysis consisting of two steps. The standard young tableaux syt of straight and skew shapes are central objects in enumerative and algebraic combinatorics. Combinatorics formula sheet factorial factorial of a nonnegative integer n. Geometric combinatorics is a branch of mathematics in general and combinatorics in particular. The following is the celebrated hook length formula. We consider permutations in this section and combinations in the next section. Catalan number is equivalent to the hook length formula. The formula involves a sum over objects called \emphexcited diagrams, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook length. Access full article top access to full text full pdf how to cite top. Perhaps the most famous example is the formula for f, the number of standard young tableaux of shape, which was discovered in 1954 by frame, robinson and thrall 22.
Combinatorics is about techniques as much as, or even more than, theorems. Abello, james monedero, a study of an independent system arising in group choice via the weak bruhat order 1985, s. Sep 28, 2010 read on an identity of glass and ng concerning the hook length formula, discrete mathematics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Similar in spirit to no occurrence obstructions in geometric complexity. In a series of papers, morales, pak, and panova prove the naruse hook length formula as well as qanalogues of naruses formula. No prior knowledge of combinatorics or representation theory is expected. Hook formula for skew shapes a paper submitted to the. Perhaps the most familiar examples are polytopes and simplicial complexes, but the subject is much broader. A simple proof of the hook length formula request pdf. Recently, naruse found a hook length formula for the number of skew shaped standard young tableaux. On an identity of glass and ng concerning the hook length. Proof of the hook length formula based on a random hook walk. The classical hook length formula hlf for the number of standard young tableaux syt of a young diagram, is a beautiful result in enumerative combinatorics that is both mysterious and extremely well studied.
In this note we present an algebraic proof of their identity. We use this result to discuss how one can arrive at a general formula for the number of young tableaux of size n. On an identity of glass and ng concerning the hook length formula. In a way it is a perfect formula highly nontrivial, clean, concise and. Geometric combinatorics mathematics mit opencourseware. Hooklength formula and applications to alternating. Hook length formula for u 2, let hu be the hook length at u, i. Based on the ideas in ckp, we introduce the weighted analogue of the branching rule for the classical hook length formula, and give two proofs of this result. Patterns in standard young tableaux and beyond math berkeley. The mathematical field of combinatorics involves determining the number of possible choices for a subset. One of the features of combinatorics is that there are usually several different ways to prove something.
Semantic scholar extracted view of hook length formula and geometric combinatorics. Hooklength formula, excited tableau, standard young tableau, flagged tableau, reverse. On an identity of glass and ng concerning the hook length formula on an identity of glass and ng concerning the hook length formula zhang, rong 20100928 00. If is a partition of nand h denotes the multiset of hook lengths of see section 3 for notation and. In this lesson, we use examples to explore the formulas that describe four combinatoric. Combinatorial geometry this is a difficult topic to define precisely without including all of discrete and computational geometry. In 2014, naruse announced a more general formula for the number. Algebraic, enumerative, probabilistic and geometric combinatorics, random walks, probabilistic group theory. C n is the number of ways that the vertices of a convex 2ngon can be paired so that the line segments joining paired vertices do not intersect. Geometric combinatorics describes a wide area of mathematics that is primarily the study of geometric objects and their combinatorial structure. Many proofs of this fact have been given, of varying complexity.
It includes a number of subareas such as polyhedral combinatorics the study of faces of convex polyhedra, convex geometry the study of convex sets, in particular combinatorics of their intersections, and discrete geometry, which in turn has many applications to computational geometry. By \things we mean the various combinations, permutations, subgroups, etc. Ives i met a man with seven wives every wife had seven sacks every sack had seven cats every cat had seven kits kits, cats, sacks, wives. The naruse hook length formula is a recent general formula for the number of standard young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook lengths. Logconcave and unimodal sequences in algebra, combinatorics, and geometry article in annals of the new york academy of sciences 5761.
Classi cation consider tilings of the 4 4 board with dominoes. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. It has applications in diverse areas such as representation theory, probability, and algorithm analysis. Geometric combinatorics in optimization and mathematical economics it is wellknown that the combinatorics of convex sets and polyhedra is extremely relevant for algorithms and. We present a transparent proof of the classical hook length formula. Combinatorics if we look at the last column, where all the permutations start with \4, we see that if we strip o. Fischera bijective proof of the hooklength formula for shifted standard tableaux.
Recently, naruse found a hooklength formula for the number of skew shaped standard. Request pdf on oct 1, 2004, kenneth glass and others published a simple proof of the hook length. The formula is reduced to an equality between the number of integer point in certain polytopes. The celebrated hook length formula gives a product formula for the number of standard young tableaux of a straight shape. The hook length formula for the number of standard young tableaux of a young diagram is a staple result in enumerative combinatorics, and as such has been widely studied. Algebraic and geometric combinatorics on lattice polytopes. Permutations of objects with some alike suppose given a collection of n objects containing k subsets of objects in which the objects in each subset are identical and objects in di erent subsets are not identical.
The formula we give will depend on the eigenvalues of ag. Request pdf on oct 1, 2004, kenneth glass and others published a simple proof of the hook length formula find, read and cite all the research you need on researchgate. The classical hook length formula gives a short product formula for the dimensions of irreducible representations of the symmetric group, and is a fundamental result in algebraic combinatorics. A probabilistic proof of a formula for the number of young tableaux of a given shape pdf.
Advanced tools such as discrete morse theory, and gromovstyle metric geometry on complexes, are also starting to take a prominent place in topological combinatorics. Introduction to hook length formula mehtaab sawhney july 22, 2016 abstract usamo 2016 problem 2 provided controversy on the nature of hook length formula hlf and whether it was \elementary. The number of standard young tableaux is given by the hook length formula of frame, robinson, and thrall. Greene, curtis, albert nijenhuis, and herbert wilf.
521 48 65 438 1287 1240 1250 487 679 969 538 554 1520 1002 751 1094 127 330 1137 1293 1506 1148 282 1371 651 431 1424 340 128 457