pinecone/core-2020-05-10-deduplication huggingface dataset:
Intersection of Isomorphic Linear Codes

AbstractGiven an (n,k) linear code C over GF(q), the intersection of C with a codeπ(C), whereπ∈Sn, is an (n,k1) code, where max{0,2k−n}⩽k1⩽k. The intersection problem is to determine which integers in this range are attainable for a given code C. We show that, depending on the structure of the generator matrix of the code, some of the values in this range are attainable. As a consequence we give a complete solution to the intersection problem for most of the interesting linear codes, e.g. cyclic codes, Reed–Muller codes, and most MDS codes
On the Number of Blocks in a Generalized Steiner System

AbstractWe considert-designs withλ=1 (generalized Steiner systems) for which the block size is not necessarily constant. An inequality for the number of blocks is derived. Fort=2, this inequality is the well known De Bruijn–Erdős inequality. Fort>2 it has the same order of magnitude as the Wilson–Petrenjuk inequality for Steiner systems with constant block size. The point of this note is that the inequality is very easy to derive and does not seem to be known. A stronger inequality was derived in 1969 by Woodall (J. London Math. Soc.(2)1, 509–519), but it requires Lagrange multipliers in the proof
Generalizations of theq-Morris Constant Term Identity

AbstractTheq-Morris constant term identity gives the constant term in the Laurent polynomial expansion ofΦ≔∏Nl=1(wl;q)a(q/wl;q)b∏1⩽j<k⩽N(wj/wk;q)λ(qwk/wj;q)λ. A conjecture is given for the constant term ofΦwhen multiplied by ∏N0+1⩽j<k⩽N(1−qλ(wj/wk))(1−qλ+1(wk/wj)). This conjecture is proved in the casesa=λ(generalN0,N1,b,λ),a=b=0 (generalN0,N1,λ), andN1=2 (generala,b,λ,N0), whereN0+N1=N. Also, a general identity relatingq-Morris-type constant terms andq-Selberg-type integrals is derived and is used to rewrite the conjecture as aq-Selberg-type integral evaluation
An Interpretation for Garsia and Remmel'sq-Hit Numbers

AbstractWe study Garsia and Remmel'sq-analogue of the rook numbers and hit numbers associated with a Ferrers board. They define theirq-rook numbers directly, and they give a three-part recurrence for the correspondingq-hit numbers. We find a new recurrence for theseq-hit numbers, and we define a statisticξon permutations that constitutes a direct combinatorial interpretation for them. The distribution ofξis invariant under column permutations of the board. For “staircase” boards, the distribution ofξis Eulerian–Mahonian. We use the statistic to prove a reciprocity theorem, and we give a new proof that the sequence of coefficents in theq-hit numbers is symmetric
Multivariable Lagrange Inversion, Gessel-Viennot Cancellation, and the Matrix Tree Theorem

AbstractA new form of multivariable Lagrange inversion is given, with determinants occurring on both sides of the equality. These determinants are principal minors, for complementary subsets of row and column indices, of two determinants that arise singly in the best known forms of multivariable Lagrange inversion. A combinatorial proof is given by considering functional digraphs, in which one of the principal minors is interpreted as a Matrix Tree determinant, and the other by a form of Gessel-Viennot cancellation
Generalized Split Graphs and Ramsey Numbers

AbstractA graphGis called a (p,q)-split graph if its vertex set can be partitioned intoA,Bso that the order of the largest independent set inAis at mostpand the order of the largest complete subgraph inBis at mostq. Applying a well-known theorem of Erdős and Rado forΔ-systems, it is shown that for fixedp,q, (p,q)-split graphs can be characterized by excluding a finite set of forbidden subgraphs, called (p,q)-split critical graphs. The order of the largest (p,q)-split critical graph,f(p,q), relates to classical Ramsey numbersR(s,t) through the inequalities2F(F(R(p+2,q+2)))+1⩾f(p,q)⩾R(p+2,q+2)−1whereF(t) is the smallest number oft-element sets ensuring at+1-elementΔ-system. Apart fromf(1,1)=5, all values off(p,q) are unknown
Dissectingd-Cubes into Smallerd-Cubes

AbstractIn this paper, we explore the following question: Given integersdandk, is it possible to subdivide ad-dimensional cube intoksmallerd-dimensional cubes? In particular, we investigate bounds on the integerc(d) which is the smallest integer for which it is possible to subdivide thed-cube into any numberk⩾c(d) smallerd-cubes. We derive specific bounds ford⩽5, and furthermore, we investigate, for givenk, the asymptotic behavior ofc(d) for thosedsuch that gcd(2d−1,kd−1)=1. Specifically, we show that if gcd (2d−1,3d−1) thenc(d)<6dand that if gcd(2d−1,kd−1) thenc(d)=O((2k)d). Finally, we derive the general asymptotic boundc(d)=O((2d)d−1) which improves the currently known bound ofc(d)=O((2d)d)
On the Neggers–Stanley Conjecture and the Eulerian Polynomials

AbstractWe prove combinatorially that theW-polynomials of naturally labeled graded posets of rank 1 or 2 (an antichain has rank 0) are unimodal, thus providing further supporting evidence for the Neggers–Stanley conjecture. For such posets we also obtain a combinatorial proof that theW-polynomials are symmetric. Combinatorial proofs that the Eulerian polynomials are log-concave and unimodal are given and we construct a simplicial complexΔwith the property that the Hilbert function of the exterior algebra modulo the Stanley–Reisner ideal ofΔis the sequence of Eulerian numbers, thus providing a combinatorial proof of a result of Brenti
Invertible Sharplyn-Transitive Sets

AbstractWe prove various results about sharplyn-transitive sets of homeomorphisms of “nice” topological spaces like the real line and the circle. Our main results concern sharply 3-transitive setsGof homeomorphisms of the circle to itself. IfG=G−1, thenGcontains a real hyperbolic part (a set of involutions of the circle having special properties). We show that every real hyperbolic part is the hyperbolic part of a real abstract oval in the sense of Buekenhout and that every real abstract oval arises from a topological oval in a flat projective plane. This establishes a new relationship between flat Minkowski planes that admit automorphisms which are circle-symmetries and flat projective planes containing topological ovals. We also consider sharplyn-transitive sets of permutations acting on finite sets. We find that our results about flat geometries and sharplyn-transitive sets of homeomorphisms have counterparts in the finite case
Percentage-Avoiding, Northwest Shapes and Peelable Tableaux

AbstractWe prove three results for Specht and Schur modules associated tonorthwest shapesand the more general class of%-avoiding shapes. The first result (conjectured for northwest shapes in by the authors) is a generalization the Littlewood–Richardson rule, giving an explicit combinatorial description for the multiplicities of irreducibles in the Specht and Schur modules of a %-avoiding shapeD, in terms ofD-peelable tableaux. The second result gives three involutions on the set of peelable tableaux which exhibit the symmetries of these multiplicities corresponding to three natural involutive operations on the set of %-avoiding shapes. The third result gives branching rules for the Specht and Schur modules of northwest shapes. The proofs are all combinatorial, with the exception of a key step in the first result, which requires results of Magyar on configuration varieties and characters of flagged Schur modules
A Randomq, t-Hook Walk and a Sum of Pieri Coefficients

AbstractThis work deals with the identityBμ(q, t)=∑ν→μcμν(q, t), whereBμ(q, t) denotes the biexponent generator of a partitionμ. That is,Bμ(q, t)=∑s∈μqa′(s)tl′(s), witha′(s) andl′(s) the co-arm and co-leg of the lattice squaresinμ. The coefficientscμν(q, t) are closely related to certain rational functions occuring in one of the Pieri rules for the Macdonald polynomials and the symbolν→μis used to indicate that the sum is over partitionsνwhich immediately precedeμin the Young lattice. This identity has an indirect manipulatorial proof involving a number of deep identities established by Macdonald. We show here that it may be given an elementary probabilistic proof by a mechanism which emulates the Greene–Nijehuis–Wilf proof of the hook formula
How Many Squares Can a String Contain?

AbstractAll our words (strings) are over afixedalphabet. A square is a subword of the formuu=u2, whereuis a nonempty word. Two squares aredistinctif they are of different shape, not just translates of each other. A worduisprimitiveifucannot be written in the formu=vjfor somej⩾2. A squareu2withuprimitive isprimitive rooted. LetM(n) denote the maximum number of distinct squares,P(n) the maximum number of distinct primitive rooted squares in a word of length n. We prove: no position in any word can be the beginning of the rightmost occurrence of more than two squares, from which we deduceM(n)<2nfor alln>0, andP(n)=n−o(n) for infinitely manyn
A Pentagonal Number Sieve

AbstractWe prove a general “pentagonal sieve” theorem that has corollaries such as the following. First, the number of pairs of partitions of n that have no parts in common isp(n)2−p(n−1)2−p(n−2)2+p(n−5)2+p(n−7)2−….Second, if two unlabeled rooted forests of the same number of vertices are chosen i.u.a.r., then the probability that they have no common tree is .8705… . Third, iff,gare two monic polynomials of the same degree over the fieldGF(q), then the probability thatf,gare relatively prime is 1−1/q. We give explicit involutions for the pentagonal sieve theorem, generalizing earlier mappings found by Bressoud and Zeilberger
A Bijective Census of Nonseparable Planar Maps

AbstractBijections are obtained between nonseparable planar maps and two different kinds of trees: description trees and skew ternary trees. A combinatorial relation between the latter and ternary trees allows bijective enumeration and random generation of nonseparable planar maps. The involved bijections take account of the usual combinatorial parameters and give a bijective proof of formulae established by Brown and Tutte. These results, combined with a bijection due to Goulden and West, give a purely combinatorial enumeration of two-stack-sortable permutations
Decompositions of Complete Multigraphs Related to Hadamard Matrices

AbstractLet bp(μKv) be the minimum number of complete bipartite subgraphs needed to partition the edge set ofμKv, the complete multigraph withμedges between each pair of itsvvertices. Many papers have examined bp(μKv) forv⩾2μ. For eachμandvwithv⩽2μ, it is shown here that if certain Hadamard and conference matrices exist, then bp(μKv) must be one of two numbers. Also, generalizations to decompositions and covers by completes-partite subgraphs are discussed and connections to designs and codes are presented
Inseparable Orthogonal Matrices over Z2

AbstractA (0, 1)-matrixAis called orthogonal over Z2if bothAATandATAare diagonal matrices. A matrixAis called inseparable ifAcontains no zero row or zero column and there do not exist permutation matricesPandQsuch that[formula]A matrixAis said to be of type 0 ifAAT=OandATA=O. A square matrixAof ordernis said to be of type 1 ifAAT=In. It turns out that an inseparable orthogonal matrix over Z2is either of type 0 or of type 1. Letf0(m,=n) (respectively,F0(m,n)) denote the smallest (respectively, largest) number of 1's in anm×ninseparable orthogonal matrix of type 0 over Z2, andf1(n) (respectively,F1(n)) denote the smallest (respectively, largest) number of 1's in ann×ninseparable orthogonal matrix of type 1 over Z2. The formulas forf0(m,n),F0(m,n),f1(n), andF1(n) are completely determined in this paper
Monomial Flocks and Herds Containing a Monomial Oval

AbstractLet F be a flock of the quadratic cone Q:X22=X1X3, inPG(3, q),qeven, and letΠt:X0=xtX1+t1/2X2+ztX3,t∈Fq, be theqplanes defining the flock F. A flock is equivalent to a herd of ovals inPG(2, q),qeven, and to a flock generalized quadrangle of order (q2, q). We show that if the herd contains a monomial oval, this oval is the Segre oval. This implies a result on the existence of subquadranglesT2(O) in the corresponding flock generalized quadrangle. To obtain this result, we prove that ifxtandztboth are monomial functions of t, then the flock is either the linear, FTWKB-, or PayneP1flock. This latter result implies, in the even case, the classification of regular partial conical flocks, as introduced by Johnson
Characterization of Certain Minimal Rank Designs

AbstractDillon asked whether the all-1 vector is in the binary code of a square (symmetric) design with parameters (22m, 22m−1−2m−1, 22m−2−2m−1) and dimension 2m+2. In this paper we show that the answer to this question is yes. This result gives a characterization of designs with these parameters and minimal 2-rank as SDP designs. Our result also allows us to remove a hypothesis from a theorem of Dillon and Schatz relating difference sets in elementary abelian 2-groups to SDP designs. Along the way we prove results about any designs with the parameters of the residual and derived designs. One of the results deals with a divisibility property that characterizes the elliptic and hyperbolic designs
Circular Numbers andn-set Partitions

AbstractPartitions ofn-sets and their associated Bell and Stirling numbers are well-studied combinatorial entities. Less studied is the connection between these entities and the moments of a Poisson random variable. We find a natural generalization of this connection by considering the moments of the circular random variables of ordernwhich are sums ofnindependent identically distributed Poisson random variables weighted equally about the unit circle. These are shown to have relationships to partitions of an-set whose blocks are of size divisible bym. From this a rich selection of combinatorial properties analogous to those of the striling numbers are explored. This includes Dobinski-type formulas, recurrence relations, and congruential properties. Some surprises are found in that form⩾3 the circular numbers are not unimodal and have no non-trivial recurrences in (n, k) of fixed ordernwith coefficients depending onk, yet they have many analogous congruential properties
A Geometric Proof of the Gap Theorem

AbstractHollmann, Körner, and Litsyn used generalized Steiner systems to prove that it is impossible to partition ann-cube intokHamming spheres if 2<k<n+2. Furthermore, ifk=n+2, they showed the only partition of then-cube consists of a single sphere of radiusn−2 andn+1 spheres of radius 0. We give a geometric proof that this is the only nontrivial partition of ann-cube into fewer than 2p+2 spheres, wherepis the largest prime withp⩽n. We also show thatk=8 is the only value ofkbetween 4 and 11 such that it is possible to partition a cube other than the (k−2)-cube intokspheres
Reconstructing Subsets of Zn

AbstractIn this paper we consider the problem of reconstructing a subsetA⊂Zn, up to translation, from the collection of its subsets of sizek, given up to translation (itsk-deck). Results of Alon, Caro, Krasikov, and Roditty (1989,J. Combin. Theory Ser. B47, 153–161) show that this is possible providedk>log2n. Mnukhin (1992,Acta. Appl. Math.29, 83–117) showed that every subset of Znof sizekis reconstructible from its (k−1)-deck, providedk⩾4. We show that whennis prime every subset of Znis reconstructible from its 3-deck; that for arbitrarynalmost all subsets of Znn are reconstructible from their 3-decks; and that for anynevery subset of Znis reconstructible from its 9α(n)-deck, whereα(n) is the number of distinct prime factors ofn. We also comment on analogous questions for arbitrary groups, in particular the cube Zn2. Our approach is to generalize the problem to that of reconstructing arbitrary rational functions on Zn. We prove—by analysing the interaction between the ideal structure of the group ring QZnand the operation of pointwise multiplication of functions—that with a suitable definition of deck every rational-valued function on Znis reconstructible from its 9α(n)-deck
A Bijection for Partitions with All Ranks at Least t

AbstractIt follows from the work of Andrews and Bressoud that fort⩽1, the number of partitions ofnwith all successive ranks at leasttis equal to the number of partitions ofnwith no part of size 2−t. We give a simple bijection for this identity which generalizes a result of Cheema and Gordon for 2-rowed plane partitions. The bijection yields several refinements of the identity when the partition counts are parametrized by the number of parts and/or the size of the Durfee rectangle. In addition, it gives an interpretation of the difference of (shifted) successive Gaussian polynomials which we relate to other interpretations of Andrews and Fishel
Rook Theory andt-Cores

AbstractIftis a positive integer, then a partition of a non-negative integernis at-core if none of the hook numbers of the associated Ferrers–Young diagram is a multiple oft. These partitions arise in the representation theory of finite groups and also in the theory of class numbers. We prove that ift=2, 3, or 4, then two differentt-cores are rook equivalent if and only if they are conjugates. In the special case whent=4, sincec4(n)=12h(−32n−20) when 8n+5 is square-free, the above result suggests a new method of approaching Gauss' class number problem for these discriminants. Unlike the cases where 2⩽t⩽4, it turns out that whent⩾5 there are distinct rook equivalentt-cores which are not conjugates. In fact, we conjecture that for all sucht, there exists a constantN(t) for which every integern⩾N(t) has the property that there exists a pair of distinct rook equivalentt-cores ofnwhich are not conjugates
Optimal Linear Perfect Hash Families

AbstractLetVbe a set of ordernand letFbe a set of orderq. A setS⊆{φ:V→F} of functions fromVtoFis an (n,q,t)-perfect hash familyif for allX⊆Vwith |X|=t, there existsφ∈Swhich is injective when restricted toX. Perfect hash families arise in compiler design, in circuit complexity theory and in cryptography. LetSbe an (n,q,t)-perfect hash family. The paper provides lower bounds on |S|, which better previously known lower bounds for many parameter sets. The paper exhibits new classes of perfect hash families which show that these lower bounds are realistic
On the Number of Solutions of a Linear Equation over Finite Sets

AbstractThe largest possible number of representations of an integer in thek-fold sumsetkA=A+…+Ais maximal forAbeing an arithmetic progression. More generally, consider the number of solutions of the linear equationc1a1+…+ckak=λ, whereci≠0 andλare fixed integer coefficients, and where the variablesairange over finite sets of integersA1,…,Ak. We prove that for fixed cardinalitiesni=|Ai|, this number of solutions is maximal whenc1=…=ck=1,λ=0 and theAiare arithmetic progressions balanced around 0 and with the same common difference. For the corresponding residues problem, assumingci,λ∈FpandAi⊆Fp(where Fpis the set of residues modulo primep), the number of solutions of the equation above does not exceed1pn1…nk+8πn1…nkn21plus;…+n2k(1+o(1))ask→∞ and under some mild restrictions onni. This is best possible save for the constant in the second term: we conjecture that in fact 8 can be replaced by 6
On the Distribution of the Area Enclosed by a Random Walk onZ2

AbstractLetΓ2nbe the set of paths with 2nsteps of unit length inZ2, which begin and end at (0,0). Forγ∈Γ2n, letarea(γ)∈Zdenote the oriented area enclosed byγ. We show that for every positive even integerk, there exists a rational functionRkwith integer coefficients, such that:1|Γ2n|∑γ∈Gamma;2n[area(γ)]k=Rk(n,n>2k.We calculate explicitly the degree and leading coefficient ofRk. We show how as a consequence of this (and by also using the enumeration of up-down permutations, and the exponential formula for cycles of permutations) one can derive the asymptotic distribution of the area enclosed by a random path inΓ2n. The formula for the asymptotic distribution can be stated as follows: forα<βinRlimn→∞|{γ∈Γ2n|nα<area(γ)<nβ}||Γ2n|=12(tanh(πβ)−tanh(πα)).Since the appropriately re-normalized random walks withnsteps onZ2converge, asn→∞, to the 2-dimensional Brownian motion, this argument can be viewed as a combinatorial approach to the formula of Lévy for the area enclosed by a random 2-dimensional Brownian path
The Veronese Surface in PG(5,3) and Witt's 5-(12,6,1) Design

AbstractA conic of the Veronese surface in PG(5,3) is a quadrangle. If one such quadrangle is replaced with its diagonal triangle, then one obtains a point model K for Witt's 5-(12,6,1) design, the blocks being the hyperplane sections containing more than three (actually six) points of K. As such a point model is projectively unique, the present construction yields an easy coordinate-free approach to some results obtained independently by Coxeter and Pellegrino, including a projective representation of the Mathieu groupM12in PG(5,3)
A Note on Difference Sets

AbstractLetDbe a (v,k,λ)-difference set in a groupG. Assume thatGhas a normal subgroupNsuch thatG/Nis cyclic or nearly cyclic. Under the self-conjugacy assumption on exp(G/N), we shall give bounds on |N| andλ. The theorem is applicable to a wider variety of parameters for groups, not necessarily abelian. These bounds exclude a (96, 20, 4)-difference set inZ/4Z×Z/8Z×Z/3ZorZ/2Z×Z/2Z×Z/8Z×Z/3Z, which were recently proved by Arasuet al. [1996, K. T. Arasu, J. A. Davis, J. Jedwab, S. L. Ma, and R. L. McFarland,in“Exponent Bounds for a Family of Abelian Difference Sets” (K. T. Arasu, J. F. Dillon, K. Harada, S. K. Sehgal, and R. L. Solomon, Eds.), pp. 129–143. DeGruyter Verlag, Berlin/New York]
How Far Can Nim in Disguise Be Stretched?

AbstractA move in the game of nim consists of taking any positive number of tokens from asinglepile. Suppose we add the class of moves of taking a nonnegative number of tokens jointly fromallthe piles. We give a complete answer to the question which moves in the class can be adjoined without changing the winning strategy of nim. The results apply to other combinatorial games with unbounded Sprague–Grundy function values. We formulate two weakened conditions of the notion of nim-sum 0 for proving the results
Inequalities for the Crank

AbstractGarvan first defined certain “vector partitions” and assigned to each such partition a “rank.” Denoting byNV(r,m,n) the (weighted) count of the vector partitions ofnwith rankrmodulom, he gave a number of relations between the numbersNV(r,m,mn+k) whenm=5, 7 and 11, 0⩽r,k<m. The true crank whose existence was conjectured by Dyson was discovered by Andrews and Garvan who also showed thatNV(r,m,n)=M(r,m,n) unlessn=1, whereM(r,m,n) denotes the number of partitions ofnwhose cranks are congruent tormodulem. In the case of module 11, a simpler form of Garvan's results have been found by Hirschhorn. In fact, the Hirschhorn result was derived using Winquist's identity, but the details were omitted. In this work, from the simpler form we deduce some new inequalities between theM(r,11,11n+k)'s and give the details of Hirschhorn's result. We also prove some conjectures of Garvan in the case of module 7
Eulerian 2-Strata Spaces

AbstractIn (B. F. Chen and M. Yan, 1998, Eulerian stratification of polyhedra,Adv. in Appl. Math.21, 22–57), we introduced the notion of Eulerian stratified spaces. In this paper, we study in detail the topological and combinatorial properties of Eulerian 2-strata spaces, as the first step toward a deeper understanding of more strata cases. We show that Eulerian 2-strata spaces have similar topological structure as topological 2-strata spaces. Moreover, we find all the linear conditions onf-vectors over arbitrary abelian group