Bounds on the number of autotopisms and subsquares of a Latin square

A subsquare of a Latin square L is a submatrix that is also a Latin square. An autotopism of L is a triplet of permutations (α, β, γ) such that L is unchanged after the rows are permuted by α, the columns are permuted by β and the symbols are permuted by γ. Let n!(n?1)!R _n be the number of n×n Latin squares. We show that an n×n Latin square has at most n ^{O(log k)} subsquares of order k and admits at most n ^{O(log n)} autotopisms. This enables us to show that {ie11-1} divides R _n for all primes p. We also extend a theorem by McKay and Wanless that gave a factorial divisor of R _n , and give a new proof that R _p ≠1 (mod p) for prime p.     

Generalized Latin squares and their defining sets

A generalized Latin square of type (n,k) is an n×n array of symbols 1,2,…,k such that each of these symbols occurs at most once in each row and each column. Let d(n,k) denote the cardinality of the minimal set S of given entries of an n×n array such that there exists a unique extension of S to a generalized Latin square of type (n,k). In this paper we discuss the properties of d(n,k) for k=2n-1 and k=2n-2. We give an alternate proof of the identity d(n,2n-1)=n²-n, which holds for even n, and we establish the new result . We also show that the latter bound is tight for n divisible by 10.     

On n-Hamiltonian line graphs

With each nonempty graph G one can associate a graph L(G), called the line graph of G, with the property that there exists a one-to-one correspondence between E(G) and V(L(G)) such that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. For integers m ≥ 2, the mth iterated line graph L^m(G) of G is defined to be L(L^m-1(G)). A graph G of order p ≥ 3 is n-Hamiltonian, 0 ≤ n ≤ p ? 3, if the removal of any k vertices, 0 ≤ k ≤ n, results in a Hamiltonian graph. It is shown that if G is a connected graph with δ(G) ≥ 3, where δ(G) denotes the minimum degree of G, then L²(G) is (δ(G) ? 3)-Hamiltonian. Furthermore, if G is 2-connected and δ(G) ≥ 4, then L²(G) is (2δ(G) ? 4)-Hamiltonian. For a connected graph G which is neither a path, a cycle, nor the graph K(1, 3) and for any positive integer n, the existence of an integer k such that L^m(G) is n-Hamiltonian for every m ≥ k is exhibited. Then, for the special case n = 1, bounds on (and, in some cases, the exact value of) the smallest such integer k are determined for various classes of graphs.     

Doubly diagonal orthogonal latin squares

A pair of doubly diagonal orthogonal latin squares of order n, DDOLS(n), is a pair of orthogonal latin squares of order n with the property that each square has a transversal on both the front diagonal (the cells {(i, i):1?i?n}) and the back diagonal (the cells {(i, n + 1?i): 1?i?n}). We show that for all n except n = 2, 3, 6, 10, 12, 14, 15, 18 and 26, there exists a pair of DDOLS(n). Obbviously these do not exist when n = 2, 3 and 6.     

Existence of weakly pandiagonal orthogonal Latin squares

A weakly pandiagonal Latin square of order n over the number set {0, 1, . . . , n-1} is a Latin square having the property that the sum of the n numbers in each of 2n diagonals is the same. In this paper, we shall prove that a pair of orthogonal weakly pandiagonal Latin squares of order n exists if and only if n ≡ 0, 1, 3 (mod 4) and n≠3.     

The intricacy of avoiding arrays is 2

Let A be any n×n array on the symbols [n], with at most one symbol in each cell. An n×n Latin square L avoids A if all entries in L differ from the corresponding entries in A. If A is split into two arrays B and C in a special way, there are Latin squares L_B and L_C avoiding B and C, respectively. In other words, the intricacy of avoiding arrays is 2, the number of arrays into which A has to be split.     

Latin parallelepipeds and cubes

For all n = 2^k, k ? 3, there are infinitely many Latin (n × n × (n ? 2))-parallelepipeds that cannot be embedded in any Latin cube of order n.     

On the mean square average of special values of L-functions

Let χ be a Dirichlet character and L(s,χ) be its L-function. Using weighted averages of Gauss and Ramanujan sums, we find exact formulas involving Jordan?s and Euler?s totient function for the mean square average of L(1,χ) when χ ranges over all odd characters modulo k and L(2,χ) when χ ranges over all even characters modulo k. In principle, using our method, it is always possible to find the mean square average of L(r,χ) if χ and r?1 have the same parity and χ ranges over all odd (or even) characters modulo k, though the required calculations become formidable when r?3. Consequently, we see that for almost all odd characters modulo k, |L(1,χ)|<Φ(k), where Φ(x) is any function monotonically tending to infinity.     

More on the generalized Macaulay theorem

Let k₁ ? k₂ ? … ? k_n be given positive integers and let F denote the set of vectors (l₁, …, l_n) with integer components satisfying 0 ? l_i ? k_i, i = 1, 2, …, n. If H is a subset of F, let (l)H denote the subset of H consisting of those vectors with component sum l, and let C((l)H) denote the smallest [(l)H] elements of (l)F. The generalized Macaulay theorem due to the author and B. Lindström [3] shows that |Gamma;((C)(l)(H)|, ? |Γ(C((l)H))|, where Γ((l)H) is the setof vectors in F obtainable by subtracting l from a single component of a vector in (l)H. A method is given for computing [Γ(C((l)H)] in this paper. It is analogous to the method for computing |Γ(C(l)H))| in the k₁ = … = k_n = 1 case which has been given independently by Katona [4] and Kruskal [5].     

Rigidity theorems for hypersurfaces with constant mean curvature

Let M ⁿ be a compact oriented hypersurface of a unit sphere \(\mathbb{S}^{n + 1} \) (1) with constant mean curvature H. Given an integer k between 2 and n ? 1, we introduce a tensor ? related to H and to the second fundamental form A of M, and show that if |?|² ≤ B _H,k and tr(? ³) ≤ C _n,k |?|³, where B _H,k and C _n,k are numbers depending only on H, n and k, then either |?|² ≡ 0 or |?|² ≡ B _H,k . We characterize all M ⁿ with |?|² ≡ B _H,k . We also prove that if \(\left| A \right|^2 \leqslant 2\sqrt {k(n - k)}\) and tr(? ³) ≤ C _n,k |?|³ then |A|² is constant and characterize all M ⁿ with |A|² in the interval \(\left[ {0,2\sqrt {k\left( {n - k} \right)} } \right] \) . We also study the behavior of |?|², with the condition additional tr(? ³) ≤ C _n,k |?|³, for complete hypersurfaces with constant mean curvature immersed in space forms and show that if sup _M |?|² = B _H,k and this supremum is attained in M ⁿ then M ⁿ is an isoparametric hypersurface with two distinct principal curvatures of multiplicities k y n ? k. Finally, we use rotation hypersurfaces to show that the condition on the trace of ? ³ is necessary in our results; more precisely, for each integer k with 2 ≤ k ≤ n ? 1 and \(H \geqslant 1/\sqrt {2n - 1} \) there is a complete hypersurface M ⁿ in \(\mathbb{S}^{n + 1} \) (1) with constant mean curvature H such that sup _M |?|² = B _H,k , and this supremum is attained in M ⁿ , and which is not a product of spheres.     

On the number of combinations without unit separation

Let f(n, k) denote the number of ways of selecting k objects from n objects arrayed in a line with no two selected having unit separation (i.e., having exactly one object between them). Then, if $\text{n ? 2(k ? 1), f(n,k)=}\text{∑}\text{i=0}\text{κ}\text{(}\text{n?k+I?2i}\text{k?2i}\text{)}$ (where $\text{κ = [}\text{k}\text{2}\text{]}$). If n < 2(k ? 1), then f(n, k) = 0. In addition, f(n, k) satisfies the recurrence relation f(n, k) = f(n ? 1, k) + f(n ? 3, k ? 1) + f(n ? 4, k ? 2). If the objects are arrayed in a circle, and the corresponding number is denoted by g(n, k), then for n > 3, g(n, k) = f(n ? 2, k) + 2f(n ? 5, k ? 1) + 3f(n ? 6, k ? 2). In particular, if n ? 2k + 1 then $\text{(n,k)=(}\text{n?k}\text{k}\text{)+(}\text{n?k?1}\text{k?1}\text{)}$.     

On avoiding odd partial Latin squares and r-multi Latin squares

We show that for any positive integer k?4, if R is a (2k-1)×(2k-1) partial Latin square, then R is avoidable given that R contains an empty row, thus extending a theorem of Chetwynd and Rhodes. We also present the idea of avoidability in the setting of partial r-multi Latin squares, and give some partial fillings which are avoidable. In particular, we show that if R contains at most nr/2 symbols and if there is an n×n Latin square L such that δn of the symbols in L cover the filled cells in R where 0<δ<1, then R is avoidable provided r is large enough.     

On the separation power and the completion of partial latin squares

Let n and m be natural numbers, n ? m. The separation power of order n and degree m is the largest integer k = k(n, m) such that for every (0, 1)-matrix A of order n with constant linesums equal to m and any set of k 1's in A there exist (disjoint) permutation matrices P₁,…, P_m such that A = P₁ + … + P_m and each of the k 1's lies in a different P_i. Almost immediately we have 1 ? k(n, m) ? m ? 1, yet in all cases where the value of k(n, m) is actually known it equals m ? 1 (except under the somewhat trivial circumstances of k(n, m) = 1). This leads to a conjecture about the separation power, namely that k(n, m) = m ? 1 if $\text{m ? [}\text{n}\text{2}\text{] + 1}$. We obtain the bound $\text{k(n, m) ? m ? [}\text{n}\text{2}\text{] + 2}$, so that this conjecture holds for n ? 7. We then move on to latin squares, describing several equivalent formulations of the concept. After establishing a sufficient condition for the completion of a partial latin square in terms of the separation power, we can show that the Evans conjecture follows from this conjecture about the separation power. Finally the lower bound on k(n, m) allows us to show, after some calculations, that the Evans conjecture is true for orders n ? 11.     

Unique reducibility of multiple blocking sets

A weighted t-fold (n ? k)-blocking set B of PG(n, q) always contains a minimal weighted t-fold (n ? k)-blocking set. We prove that, if ${|B| < (t+1)q^{n-k} + \theta_{n-k-1}}$ , then the minimal weighted t-fold (n ? k)-blocking set contained in B is unique.     

Embedding finite posets in cubes

In this paper we define the n-cube Q_n as the poset obtained by taking the cartesian product of n chains each consisting of two points. For a finite poset X, we then define dim₂X as the smallest positive integer n such that X can be embedded as a subposet of Q_n. For any poset X we then have log₂ |X| ? dim₂X ? |X|. For the distributive lattice $\text{L =}\text{2}{}^{\mathrm{X}}$, dim₂L = |X| and for the crown S^k_n, dim₂ (S^k_n) = n + k. For each k ? 2, there exist positive constants c₁ and c₂ so that for the poset X consisting of all one element and k-element subsets of an n-element set, the inequality c₁ log₂n < dim₂(X) < c₂ log₂n holds for all n with k < n. A poset is called Q-critical if dim₂ (X ? x) < dim₂(X) for every x ? X. We define a join operation ⊕ on posets under which the collection Q of all Q-critical posets which are not chains forms a semigroup in which unique factorization holds. We then completely determine the subcollection $\text{M}$ ? Q consisting of all posets X for which dim₂ (X) = |X|.     

Successes,runs and longest runs

The probability distribution of the number of success runs of length k (⩾1) in n (⩾1) Bernoulli trials is obtained. It is noted that this distribution is a binomial distribution of order k, and several open problems pertaining to it are stated. Let S_n and L_n, respectively, denote the number of successes and the length of the longest success run in the n Bernoulli trials. A formula is derived for the probability P(L_n ⩽ k | S_n = r) (0 ⩽ k ⩽ r ⩽ n), which is alternative to those given by Burr and Cane (1961) and Gibbons (1971). Finally, the probability distribution of X_{n, Ln}^(k) is established, where X_{n, Ln}^(k) denotes the number of times in the n Bernoulli trials that the length of the longest success run is equal to k.     

Successes,runs and longest runs

The probability distribution of the numbeer of success runs of length k ( >/ 1) in n ( ⩾ 1) Bernoulli trials is obtained. It is noted that this distribution is a binomial distribution of order k, and several open problems pertaining to it are stated. Let S_n and L_n, respectively, denote the number of successes and the length of the longest success run in the n Bernoulli trials. A formula is derived for the probability P(L_n ⩽ k | S_n = r) (0 ⩽ k ⩽ r ⩽ n), which is alternative to those given by Burr and Cane (1961) and Gibbons (1971). Finally, the probability distribution of X_{n, Ln}^(k) is established, where X_{n, Ln}^(k) denotes the number of times in the n Bernoulli trials that the length of the longest success run is equal to k.     

The number of meets between two subsets of a lattice

Let L be a lattice of divisors of an integer (isomorphically, a direct product of chains). We prove |A| |B| ? |L| |A ∧ B| for any A, B ? L, where |·| denotes cardinality and A ∧ B = {a ∧ b: a?A, b?B}. |A ∧ B| attains its minimum for fixed |A|, |B| when A and B are ideals. |·| can be replaced by certain other weight functions. When the n chains are of equal size k, the elements may be viewed as n-digit k-ary numbers. Then for fixed |A|, |B|, |A ∧ B| is minimized when A and B are the |A| and |B| smallest n-digit k-ary numbers written backwards and forwards, respectively. |A ∧ B| for these sets is determined and bounded. Related results are given, and conjectures are made.     

Second moments of Dirichlet L-functions weighted by Kloosterman sums

For the general modulo q ? 3 and a general multiplicative character χ modulo q, the upper bound estimate of |S(m, n, 1, χ, q)| is a very complex and difficult problem. In most cases, the Weil type bound for |S(m, n, 1, χ, q)| is valid, but there are some counterexamples. Although the value distribution of |S(m, n, 1, χ, q)| is very complicated, it also exhibits many good distribution properties in some number theory problems. The main purpose of this paper is using the estimate for k-th Kloosterman sums and analytic method to study the asymptotic properties of the mean square value of Dirichlet L-functions weighted by Kloosterman sums, and give an interesting mean value formula for it, which extends the result in reference of W. Zhang, Y.Yi, X.He: On the 2k-th power mean of Dirichlet L-functions with the weight of general Kloosterman sums, Journal of Number Theory, 84 (2000), 199–213.     

Triangulations of orientable surfaces by complete tripartite graphs

Orientable triangular embeddings of the complete tripartite graph K_n,n,n correspond to biembeddings of Latin squares. We show that if n is prime there are at least e^{nlnn-n(1+o(1))} nonisomorphic biembeddings of cyclic Latin squares of order n. If n=kp, where p is a large prime number, then the number of nonisomorphic biembeddings of cyclic Latin squares of order n is at least e^{plnp-p(1+lnk+o(1))}. Moreover, we prove that for every n there is a unique regular triangular embedding of K_n,n,n in an orientable surface.