Maximizing a correlational ratio for linear extensions of posets

Let ρ=P(12)/P(12|13), where P(ij) is the probability that i precedes j in a randomly chosen linear extension of a partially ordered set ({1,2,...,n}, ≤) in which points 1, 2 and 3 are mutually incomparable. A previous paper by the author (Order 1, 127 (1984)) proved that ρ≤1. The present paper considers the maximization of ρ for each n≥3. It shows that, with \(\alpha _n = \left\lfloor {(n + 3)/2} \right\rfloor \) , the maximum ρ is at least $$\left[ {\alpha _n \left( {\begin{array}{*{20}c} n \\ {\alpha _n } \\ \end{array} } \right) - n} \right]/\left[ {\alpha _n \left( {\begin{array}{*{20}c} n \\ {\alpha _n } \\ \end{array} } \right) - \alpha _n } \right]$$ . Evidence that this value cannot be exceeded is given. It is also proved that the smallest possible value of P(231)+P(321) is $$1/\left( {\begin{array}{*{20}c} n \\ {\left\lfloor {\left( {n + 1} \right)/2} \right\rfloor } \\ \end{array} } \right)$$ .     

On some new identities for the Fibonomial coefficients

Let F _n be the nth Fibonacci number. The Fibonomial coefficients \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F\) are defined for n ≥ k > 0 as follows $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = \frac{{F_n F_{n - 1} \cdots F_{n - k + 1} }} {{F_1 F_2 \cdots F_k }},$$ with \(\left[ {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right]_F = 1\) and \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = 0\) . In this paper, we shall provide several identities among Fibonomial coefficients. In particular, we prove that $$\sum\limits_{j = 0}^{4l + 1} {\operatorname{sgn} (2l - j)\left[ {\begin{array}{*{20}c} {4l + 1} \\ j \\ \end{array} } \right]_F F_{n - j} = \frac{{F_{2l - 1} }} {{F_{4l + 1} }}\left[ {\begin{array}{*{20}c} {4l + 1} \\ {2l} \\ \end{array} } \right]_F F_{n - 4l - 1} ,}$$ holds for all non-negative integers n and l.     

An explicit formula for |FM(1+1+n)|

We show that the number of elements in FM(1+1+n), the modular lattice freely generated by two single elements and an n-element chain, is 1 $$\frac{1}{{6\sqrt 2 }}\sum\limits_{k = 0}^{n + 1} {\left[ {2\left( {\begin{array}{*{20}c} {2k} \\ k \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {2k} \\ {k - 2} \\ \end{array} } \right)} \right]} \left( {\lambda _1^{n - k + 2} - \lambda _2^{n - k + 2} } \right) - 2$$ , where \(\lambda _{1,2} = {{\left( {4 \pm 3\sqrt 2 } \right)} \mathord{\left/ {\vphantom {{\left( {4 \pm 3\sqrt 2 } \right)} 2}} \right. \kern-0em} 2}\) .     

On the normalizing multiplier of the generalized Jackson kernel

We consider the question of evaluating the normalizing multiplier $$\gamma _{n,k} = \frac{1}{\pi }\int_{ - \pi }^\pi {\left( {\frac{{sin\tfrac{{nt}}{2}}}{{sin\tfrac{t}{2}}}} \right)^{2k} dt} $$ for the generalized Jackson kernel J _n,k (t). We obtain the explicit formula $$\gamma _{n,k} = 2\sum\limits_{p = 0}^{\left[ {k - \tfrac{k}{n}} \right]} {( - 1)\left( {\begin{array}{*{20}c} {2k} \\ p \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {k(n + 1) - np - 1} \\ {k(n - 1) - np} \\ \end{array} } \right)} $$ and the representation $$\gamma _{n,k} = \sqrt {\frac{{24}}{\pi }} \cdot \frac{{(n - 1)^{2k - 1} }}{{\sqrt {2k - 1} }}\left[ {1\frac{1}{8} \cdot \frac{1}{{2k - 1}} + \omega (n,k)} \right],$$ , where $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ .     

The number of strictly increasing mappings of fences

Let X and Y be fences of size n and m, respectively and n, m be either both even or both odd integers (i.e., |m-n| is an even integer). Let \(r = \left\lfloor {{{(n - 1)} \mathord{\left/ {\vphantom {{(n - 1)} 2}} \right. \kern-0em} 2}} \right\rfloor\) . If 1<n<-m then there are \(a_{n,m} = (m + 1)2^{n - 2} - 2(n - 1)(\begin{array}{*{20}c} {n - 2} \\ r \\ \end{array} )\) of strictly increasing mappings of X to Y. If 1<-m<-n<-2m and s=1/2(n?m) then there are a _n,m+b _n,m+c _n of such mappings, where $$\begin{gathered} b_{n,m} = 8\sum\limits_{i = 0}^{s - 2} {\left( {\begin{array}{*{20}c} {m + 2i + 1} \\ l \\ \end{array} } \right)4^{s - 2 - 1} } \hfill \\ {\text{ }}c_n = \left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {n - 1} \\ {s - 1} \\ \end{array} } \right){\text{ if both }}n,m{\text{ are even;}} \hfill \\ {\text{ 0 if both }}n,m{\text{ are odd}}{\text{.}} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$      

Matrix semigroups—generators and relations

Some presentations for the semigroups of all 2×2 matrices and all 2×2 matrices of determinant 0 or 1 over the field GF(p) (p prime) are given. In particular, if <a, b, c‖ R> is any (semigroup) presentation for the general linear group in terms of generators $$A = \left( {\begin{array}{*{20}c} 1 & 0 \\ 1 & 1 \\ \end{array} } \right),B = \left( {\begin{array}{*{20}c} 1 & 1 \\ 0 & 1 \\ \end{array} } \right),C = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & \xi \\ \end{array} } \right),$$ where ζ is a primitive root of 1 modulop, then the presentation $$\langle a,b,c,t|R,t^2 = ct = tc = t,tba^{p - 1} t = 0,b^{\xi - 1} atb = a^{\xi - 1} tb^\xi a^{1 - \xi - 1} \rangle $$ defines the semigroup of all 2×2 matrices over GF (2,p) in terms of generatorsA, B, C and $$T = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & 0 \\ \end{array} } \right).$$ Generating sets and ranks of various matrix semigroups are also found.     

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

On Ramsey—Turán type theorems for hypergraphs

LetH ^r be anr-uniform hypergraph. Letg=g(n;H ^r ) be the minimal integer so that anyr-uniform hypergraph onn vertices and more thang edges contains a subgraph isomorphic toH ^r . Lete =f(n;H ^r ,εn) denote the minimal integer such that everyr-uniform hypergraph onn vertices with more thane edges and with no independent set ofεn vertices contains a subgraph isomorphic toH ^r . We show that ifr>2 andH ^r is e.g. a complete graph then $$\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} f(n;H^r ,\varepsilon n) = \mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} g(n;H^r )$$ while for someH ^r with \(\mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} g(n;H^r ) \ne 0\) $$\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} f(n;H^r ,\varepsilon n) = 0$$ . This is in strong contrast with the situation in caser=2. Some other theorems and many unsolved problems are stated.     

On cyclic compositions of positive integers

Say that two compositions of n into k parts are related if they differ only by a cyclic shift. This defines an equivalence relation on the set of such compositions. Let ${\left\langle \begin{array}{c}n \\ k\end{array} \right\rangle}$ denote the number of distinct corresponding equivalence classes, that is, the number of cyclic compositions of n into k parts. We show that the sequence ${\left\langle\begin{array}{c}n \\ k\end{array}\right\rangle}$ is log-concave and prove some results concerning ${\left\langle \begin{array}{c}n \\ k \end{array} \right\rangle}$ modulo two.     

Combinatorial Aspects of the Splitting Number

This paper deals with the splitting number ${\mathfrak{s}}$ and polarized partition relations. In the first section we define the notion of strong splitting families, and prove that its existence is equivalent to the failure of the polarized relation $$\left(\begin{array}{lll}\mathfrak{s} \\ \omega \end{array} \right) \rightarrow {\left(\begin{array}{ll}\mathfrak{s} \\ \omega \end{array} \right)}^{1, 1}_{2}$$ . We show that the existence of a strong splitting family is consistent with ZFC, and that the strong splitting number equals the splitting number, when it exists. Consequently, we can put some restriction on the possibility that s is singular. In the second section we deal with the polarized relation under the weak diamond, and we prove that the strong polarized relation $$\left(\begin{array}{lll}2^{\omega} \\ \omega \end{array} \right) \rightarrow {\left(\begin{array}{ll}2^{\omega} \\ \omega \end{array} \right)}^{1, 1}_{2}$$ is consistent with ZFC, even when cf ${(2^{\omega}) = \aleph_{1}}$ (hence the weak diamond holds).     

A decomposition of the universal embedding space for the near polygon {{\mathbb H}_n}

Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ _i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ .     

Metric theorems related to the Kronecker-Weyl theorem modm

Let (Ω, ?,P) be the infinite product of identical copies of the unit interval probability space. For a Lebesgue measurable subsetI of the unit interval, let \(A(N,I,\omega ) = \# \left\{ {n \leqslant N|\omega _n \varepsilon I} \right\}\) , where ω=(ω₁,ω₂,...). For integersm>1, and 0≤r<m, define $$\varepsilon (k,r,m,I,\omega ) = \left\{ {\begin{array}{*{20}c} {1\,if\,A(k,I,\omega ) \equiv r(\bmod m)} \\ {0\,otherwise} \\ \end{array} } \right.$$ and $$\eta (k,m,I,\omega ) = \left\{ {\begin{array}{*{20}c} {1\,if\,(A(k,I,\omega ),m) \equiv 1} \\ {0\,otherwise.} \\ \end{array} } \right.$$ A theorem ofK. L. Chung yields an iterated logarithm law and a central limit theorem for sums of the variables ε(k) and η(k).     

On the representation theory of Möbius groups inR n*

We will solve several fundamental problems of Möbius groupsM(R ⁿ) which have been matters of interest such as the conjugate classification, the establishment of a standard form without finding the fixed points and a simple discrimination method. Let \(g = \left[ {\begin{array}{*{20}c} a &; b \\ c &; d \\ \end{array} } \right]\) be a Clifford matrix of dimensionn, c ≠ 0. We give a complete conjugate classification and prove the following necessary and sufficient conditions:g is f.p.f. (fixed points free) iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; 0 \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α|<1 and |E?AE ¹| ≠ 0;g is elliptic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; \beta \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α| <1 and |E?AE ¹|=0;g is parabolic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; 0 \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α|=1; andg is loxodromic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; \beta \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α| >1 or rank (E?AE ¹) ≠ rank (E?AE ¹,ac ^?1+c ^?1 d), where α is represented by the solutions of certain linear algebraic equations and satisfies $\left| {c^{ - 1} \alpha '} \right| = \left| {\left( {E - AE^1 } \right)^{ - 1} \left( {\alpha c^{ - 1} + c^{ - 1} \alpha '} \right)} \right|.$      

On designated-weight Boolean functions with highest algebraic immunity

Algebraic immunity has been considered as one of cryptographically significant properties for Boolean functions. In this paper, we study ∑d-1 i=0 (ni)-weight Boolean functions with algebraic immunity achiev-ing the minimum of d and n - d + 1, which is highest for the functions. We present a simpler sufficient and necessary condition for these functions to achieve highest algebraic immunity. In addition, we prove that their algebraic degrees are not less than the maximum of d and n - d + 1, and for d = n1 +2 their nonlinearities equalthe minimum of ∑d-1 i=0 (ni) and ∑ d-1 i=0 (ni). Lastly, we identify two classes of such functions, one having algebraic degree of n or n-1.     

Lipschitz-Nikolski\imath constants and asymptotic simultaneous approximation on theM n -operatorsconstants and asymptotic simultaneous approximation on theM n -operators

This note is a study of approximation of classes of functions and asymptotic simultaneous approximation of functions by theM _n -operators of Meyer-König and Zeller which are defined by $$(M_n f)(x) = (1 - x)^{n + 1} \sum\limits_{k = 0}^\infty {f\left( {\frac{k}{{n + k}}} \right)} \left( \begin{array}{l} n + k \\ k \\ \end{array} \right)x^k , n = 1,2,....$$ Among other results it is proved that for 0<α≤1 $$\mathop {\lim }\limits_{n \to \infty } n^{\alpha /2} \mathop {\sup }\limits_{f \in Lip_1 \alpha } \left| {(M_n f)(x) - f(x)} \right| = \frac{{\Gamma \left( {\frac{{\alpha + 1}}{2}} \right)}}{{\pi ^{1/2} }}\left\{ {2x(1 - x)^2 } \right\}^{\alpha /2} $$ and if for a functionf, the derivativeD ^m+2 f exist at a pointx∈(0, 1), then $$\mathop {\lim }\limits_{n \to \infty } 2n[D^m (M_n f) - D^m f] = \Omega f,$$ where Ω is the linear differential operator given by $$\Omega = x(1 - x)^2 D^{m + 2} + m(3x - 1)(x - 1)D^{m + 1} + m(m - 1)(3x - 2)D^m + m(m - 1)(m - 2)D^{m - 1} .$$      

An example on strong shift equivalence of positive integral matrices

The following matrices are considered $$A_k = \left( {\begin{array}{*{20}c} k \\ 1 \\ \end{array} \begin{array}{*{20}c} 2 \\ k \\ \end{array} } \right), B_k \left( {\begin{array}{*{20}c} {k - 1} \\ 1 \\ \end{array} \begin{array}{*{20}c} 1 \\ {k + 1} \\ \end{array} } \right),k \in \mathbb{N},$$ which are strong shift equivalent in the sense ofWilliams [7]. In case \(k + \sqrt 2 \) is a prime number of the algebraic field \(\mathbb{Q}(\sqrt 2 )\) matrices are defined which determine the possible choices of rank two matrices connectingA _k andB _k in the sense of strong shift equivalence. A complete list of all these matrices is given.     

On prime numbers of special kind on short intervals

Suppose that the Riemann hypothesis holds. Suppose that $$\psi _1 (x) = \mathop \sum \limits_{\begin{array}{*{20}c} {n \leqslant x} \\ {\{ (1/2)n^{1/c} \} < 1/2} \\ \end{array} } \Lambda (n)$$ where c is a real number, 1 < c ≤ 2. We prove that, for H>N ^1/2+10ε, ε > 0, the following asymptotic formula is valid: $$\psi _1 (N + H) - \psi _1 (N) = \frac{H}{2}\left( {1 + O\left( {\frac{1}{{N^\varepsilon }}} \right)} \right)$$ .     

A reciprocity formula for quadratic forms

LetQ(x) denote a quadratic form over the rational integers in four variables (x=(x₁,...,x₄)). ThenQ is representable as a symmetric matrix. Assume this matrix to be non-singular modp(p≠2 prime); then the “inverse” quadratic formQ ^?1 modp can be defined. Letf:?⁴→? be defined such that the Fourier transformf exists and the sum $$\sum\limits_{x \in \mathbb{Z}^4 } {f(c x), c \in \mathbb{R}, c \ne 0} $$ is convergent. Furthermore, letm=p ₁...p _k be the product ofk distinct primes withm>1, 2×m; let $$\varepsilon = \prod\limits_{i = 1}^k {\left( {\frac{{\det Q}}{{p_i }}} \right)} \ne 0$$ for the Legendre symbol $$\left( {\frac{ \cdot }{p}} \right)$$ ; define $$B_i (Q,x) = \left\{ {\begin{array}{*{20}c} {1 for Q(x) \equiv 0\bmod p_i } \\ , \\ {0 for Q(x)\not \equiv 0\bmod p_i } \\ \end{array} } \right.$$ and forr∈?,r>0, $$F(Q,f,r) = \sum\limits_{x \in \mathbb{Z}^4 } {\left( {\prod\limits_{i = 1}^k {\left( {B_i (Q,x) - \frac{1}{{p_i }}} \right)} } \right)f(r^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} x)} $$ Then we have $$F(Q,f,m) = \varepsilon F(Q^{ - 1} ,\hat f,m)$$      

On sums of binomial coefficients involving Catalan and Delannoy numbers modulo $$p^2$$

In this paper, we prove some congruences conjectured by Z.-W. Sun: For any prime \(p>3\), we determine

$$\begin{aligned} \sum \limits _{k = 0}^{p - 1} {\frac{{{C_k}C_k^{(2)}}}{{{{27}^k}}}} \quad {\text { and }}\quad \sum \limits _{k = 1}^{p - 1} {\frac{{\left( {\begin{array}{l} {2k} \\ {k - 1} \\ \end{array}} \right) \left( { \begin{array}{l} {3k} \\ {k - 1} \\ \end{array} } \right) }}{{{{27}^k}}}} \end{aligned}$$

modulo \(p^2\), where \(C_k=\frac{1}{k+1}\left( {\begin{array}{c}2k\\ k\end{array}}\right) \) is the k-th Catalan number and \(C_k^{(2)}=\frac{1}{2k+1}\left( {\begin{array}{c}3k\\ k\end{array}}\right) \) is the second-order Catalan numbers of the first kind. And we prove that

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{D_k}{k}\equiv -q_p(2)+pq_p(2)^2\pmod {p^2}, \end{aligned}$$

where \(D_n=\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n+k\\ k\end{array}}\right) \) is the n-th Delannoy number and \(q_p(2)=(2^{{p-1}}-1)/p\) is the Fermat quotient.

     

Enumerating rooted simple planar maps

The main purpose of this paper is to find the number of combinatorially distinct rooted simpleplanar maps,i.e.,maps having no loops and no multi-edges,with the edge number given.We haveobtained the following results.1.The number of rooted boundary loopless planar [m,2]-maps.i.e.,maps in which there areno loops on the boundaries of the outer faces,and the edge number is m,the number of edges on theouter face boundaries is 2,is(?)for m≥1.G_0~N=0.2.The number of rooted loopless planar [m,2]-maps is(?)3.The number of rooted simple planar maps with m edges H_m~s satisfies the following recursiveformula:(?)where H_m~(NL) is the number of rooted loopless planar maps with m edges given in [2].4.In addition,γ(i,m),i≥1,are determined by(?)for m≥i.γ(i,j)=0,when i>j.