On the average genus of the random graph

The generalized Petersen graph GP (n, k), n ≤ 3, 1 ≥ k < n/2 is a cubic graph with vertex-set {u_j; i ? Z_n} ∪ {v_j; i ? Z_n}, and edge-set {u_iu_i, u_iv_i, v_iv_{i+k, i?}Z_n}. In the paper we prove that (i) GP(n, k) is a Cayley graph if and only if k² ? 1 (mod n); and (ii) GP(n, k) is a vertex-transitive graph that is not a Cayley graph if and only if k² ? -1 (mod n) or (n, k) = (10, 2), the exceptional graph being isomorphic to the 1-skeleton of the dodecahedon. The proof of (i) is based on the classification of orientable regular embeddings of the n-dipole, the graph consisting of two vertices and n parallel edges, while (ii) follows immediately from (i) and a result of R. Frucht, J.E. Graver, and M.E. Watkins [“The Groups of the Generalized Petersen Graphs,” Proceedings of the Cambridge Philosophical Society, Vol. 70 (1971), pp. 211-218]. © 1995 John Wiley & Sons, Inc.     

Permutations and successions

Let π = (π(1), π(2),…, π(n)) be a permutation on {1, 2, …, n}. A succession (respectively, ¹-succession) in π is any pair π(i), π(i + 1), where π(i + 1) = π(i) + 1 (respectively, π(i + 1) ≡ π(i) + 1 (mod n)), i = 1, 2, …, n ? 1. Let R(n, k) (respectively, $\text{R}{}^1\text{(n, k)}$) be the number of permutations with k successions (respectively, ¹-successions). In this note we determine R(n, k) and $\text{R}{}^1\text{(n, k)}$. In addition, these notions are generalized to the case of circular permutations, where analogous results are developed.     

The asymptotic ruin problem when the healthy and sick periods form an alternating renewal process

Let {T₁, Y₁}^∞_i=1 be a sequence of positive independent random variables. Let, also, Z₁ = βY′₁ ? πT_i, i = 1, 2, …, where Y′₁ = Max(0, Y_i ? w), w ? 0, and where β < 0 and π is such that E(Z₁) < 0. We consider the random walk of partial sums S_n = ?ⁿ_i=1Z_i in the presence of an absorbing region (u, ∞), u ? 0, and S₀ ≡ 0. Of interest is $\text{ψ(u) = Pr(}\text{S}\text{?}\text{≤ u)}$ where S? = Sup(0, S₁, S₂, …, S_n, …).     

A symmetry relationship for a class of partitions

Let S(n, k, v) denote the number of vectors (a₀,…, a_n?1) with nonnegative integer components that satisfy a₀ + … + a_{n ? 1} = k and Σ_i=0^n?1ia_i ≡ v (mod n). Two proofs are given for the relation S(n, k, v) = S(k, n, v). The first proof is by algebraic enumeration while the second is by combinatorial construction.     

Isomorphism of circulant graphs and digraphs

Let S? {1, …, n?1} satisfy ?S = S mod n. The circulant graph G(n, S) with vertex set {v₀, v₁,…, v_n?1} and edge set E satisfies v_iv_j?E if and only if j ? i ∈ S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = ? S. Ádám conjectured that $\text{G(n, S) ? G(n, S′)}$ if and only if S = uS′ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p², and determine exact conditions on S that it be true in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p² where p is an odd prime, or n is divisible by 2⁴.     

A class of facet producing graphs for vertex packing polyhedra

We examine a family of graphs called webs. For integers n ? 2 and $\text{k, 1 ? k ?}\text{1}\text{2}\text{n}$, the web W(n, k) has vertices V_n = {1, …, n} and edges {(i, j): j = i+k, …, i+n ? k, for i?V_n (sums mod n)}. A characterization is given for the vertex packing polyhedron of W(n, k) to contain a facet, none of whose projections is a facet for the lower dimensional vertex packing polyhedra of proper induced subgraphs of W(n, k). Simple necessary and sufficient conditions are given for W(n, k) to contain W(n′, k′) as an induced subgraph; these conditions are used to show that webs satisfy the Strong Perfect Graph Conjecture. Complements of webs are also studied and it is shown that if both a graph and its complement are webs, then the graph is either an odd hole or its complement.     

Enumeration of rises and falls by position

Let π=(π₁, π₂,…,π_n) denote a permutation of Z_n = {1, 2,…, n}. The pair (π_i, π_i+1) is a rise if π_i<π_i+1 or a fall if π_i>π_i+1. Also a conventional rise is counted at the beginning of π and a conventional fall at the end. Let k be a fixed integer ≥ 1. The rise π_i,π_i+1 is said to be in a in a j (mod k) position if i ≡ j (mod k); similarly for a fall. The conventional rise at the beginning is in a 0 (mod k) position, while the conventional fall at the end is in an n (mod k) position. Let $\text{P}{}_{\mathrm{n}}\text{≡P}{}_{\mathrm{n}}\text{(r}{}_0\text{,…,r}{}_{\mathrm{k}}\text{?1,?}{}_0\text{,…,?;}{}_{\mathrm{k}}\text{?1)}$ denote the number of permutations having r_i rises i (mod k) positions and ?;_i falls in i (mod k) positions. A generating function for P_n is obtained. In particular, for k = 2 the generating function is quite explicit and also, for certain special cases when k = 4.     

A generalization of the Kruskal-Katona theorem

Let k_n ? k_n?1 ? … ? k₁ be positive integers and let ($\text{i}\text{j}$) denote the coefficient of xⁱ in $\text{Π}\text{r=1}\text{j}\text{(1 + x + x}{}^2\text{+ … + x}{}^{\mathrm{kr}}\text{)}$. For given integers l, m, where 1 ? l ? k_n + k_n?1 + … + k₁ and $\text{1 ? m ? (}\text{n}\text{n}\text{)}$, it is shown that there exist unique integers m(l), m(l ? 1),…, m(t), satisfying certain conditions, for which $\text{m = (}\text{m(l)}\text{l}\text{+ (}\text{m(l?1)}\text{l?1}\text{) + … + (}\text{m(t)}\text{t}\text{)}$. Moreover, any m l-subsets of a multiset with k_i elements of type i, i = 1, 2,…, n, will contain at least $\text{(}\text{m(l)}\text{l?1}\text{) + (}\text{m(l?1)}\text{l?2}\text{) + … + (}\text{m(t)}\text{t?1}$ different (l ? 1)-subsets. This result has been anticipated by Greene and Kleitman, but the formulation there is not completely correct. If k₁ = 1, the numbers ($\text{j}\text{i}$) are binomial coefficients and the result is the Kruskal-Katona theorem.     

A direct method to construct triple systems

A triple system is a balanced incomplete block design D(v, k, λ, b, r) with k = 3. Although it has been shown that triple systems exist for all values of the parameters satisfying the necessary conditions:

$\text{λ(ν ? 1) ≡ 0 (}\text{mod 2}\text{), λν(ν ? 1) ≡ 0 (}\text{mod 6}\text{),}$

direct methods (nonrecursive) of construction are not available in general. In this paper we give a direct method to construct a triple system for all values of the parameters satisfying the necessary conditions.     

Hecke operators and an identity for the dedekind sums

If h, k ∈ Z, k > 0, the Dedekind sum is given by

$\text{s(h,k) =}\text{∑}\text{μ=1}\text{k}\text{μ}\text{k}\text{hμ}\text{k}$

, with

$\text{((x)) = x ? [x] ?}\text{1}\text{2}\text{, x?Z}$

,

$\text{=0 , x∈Z}$

. The Hecke operators T_n for the full modular group SL(2, Z) are applied to log η(τ) to derive the identities (n ∈ Z⁺)

$\text{∑ ∑ s(ah+bk,dk) = σ(n)s(h,k)}$

,

$\text{ad=n b(}\text{mod}\text{d)}$

$\text{d>0}$

where (h, k) = 1, k > 0 and σ(n) is the sum of the positive divisors of n. Petersson had earlier proved (1) under the additional assumption k ≡ 0, h ≡ 1 (mod n). Dedekind himself proved (1) when n is prime.     

The asymptotic distribution of weighted empirical distribution functions

Let G_n denote the empirical distribution based on n independent uniform (0, 1) random variables. The asymptotic distribution of the supremum of weighted discrepancies between G_n(u) and u of the forms 6w_v(u)D_n(u)6 and 6w_v(G_n(u))D_n(u)6, where D_n(u) = G_n(u)?u, w_v(u) = (u(1?u))^?1+v and 0 ? v < $\text{1}\text{2}$ is obtained. Goodness-of-fit tests based on these statistics are shown to be asymptotically sensitive only in the extreme tails of a distribution, which is exactly where such statistics that use a weight function w_v with $\text{1}\text{2}$ ? v ? 1 are insensitive. For this reason weighted discrepancies which use the weight function w_v with 0 ? v < $\text{1}\text{2}$ are potentially applicable in the construction of confidence contours for the extreme tails of a distribution.     

Existence and comparison theorems for nonlinear diffusion systems

In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v¹(x), v²(x),…, v^m(x)) from $\text{Ω ? R}{}^{\mathrm{n}}$ into R^m which satisfy ψⁱ(x, t) ? vⁱ(x) ? θⁱ(x, t) for all $\text{x ? Ω}\text{and}\text{1 ? i ? m}$, where ψⁱand θⁱ are extended real-valued functions on $\text{}\text{?}\text{gW × [0, T)}$. We find conditions which will ensure that a solution U(x, t) ≡ (u¹(x, t), u²(x, t),…, u^m(x, t)) which satisfies U(x, 0) ?K(0) will also satisfy U(x, t) ?K(t) for all 0 ? t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.     

An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

On maximal antichains containing no set and its complement

Let 1?k₁?k₂?…?k_n be integers and let S denote the set of all vectors x = (x₁, …, x_n with integral coordinates satisfying 0?x_i?k_i, i = 1,2, …, n; equivalently, S is the set of all subsets of a multiset consisting of k_i elements of type i, i = 1,2, …, n. A subset X of S is an antichain if and only if for any two vectors x and y in X the inequalities x_i?y_i, i = 1,2, …, n, do not all hold. For an arbitrary subset H of S, (i)H denotes the subset of H consisting of vectors with component sum i, i = 0, 1, 2, …, K, where K = k₁ + k₂ + …k_n. |H| denotes the number of vectors in H, and the complement of a vector x?S is (k₁-x₁, k₂-x₂, …, k_n -x_n). What is the maximal cardinality of an antichain containing no vector and its complement? The answer is obtained as a corollary of the following theorem: if X is an antichain, K is even and$\text{|(}\text{1}\text{2}\text{K)X|}$ does not exceed the number of vectors in $\text{(}\text{1}\text{2}\text{K)S}$ with first coordinate different from k₁, then

$\text{∑}\text{i=0}\text{K}\text{i≠}\text{1}\text{2}\text{K}\text{|(i)X|}\text{|(i)S|}\text{+}\text{|(}\text{1}\text{2}\text{K)X|}\text{|(}\text{1}\text{2}\text{K-1)S|}\text{?1}$

.     

Ballots and rooks

The level code representation of the simplest ballot problem (weak lead lattice paths from (0, 0) to (n, n) is the set of sequences (b₁,…, b_n) defined by b₁ = 1, b_{i ?1} ≤ b_i ≤ i, 2 ≤ i ≤n. Each sequence is monotone non-decreasing, has a specification (c₁, c₂,…, c_n) with c_i the number of sequence elements equal to i (hence c₁ + c₂…+ c_n = n), and may be permuted in $\text{n!}\text{c}{}_1\text{!…c}{}_{\mathrm{n}}\text{!}$ ways. The set of permuted sequences, as noted in [4], is the set of parking functions, introduced by Konheim and Weiss in [1]. To count parking functions by number of fixed points, associate the rook polynomial for matching a deck of cards of specification (c₁,…,c_n), c_i cards marked i, with a deck of n distinct cards. The hit polynomial H_n(x) corresponding to the sum of such rook polynomials over all sequences (I am using the terminology of [2]) is the required enumerator and turns out to be simply

$\text{(n+1)}{}^2\text{H}{}_{\mathrm{n}}\text{(x)=(x+n)}{}^{\mathrm{n+1}}\text{?(x?1)}{}^{\mathrm{n+1}}$

.     

Decomposable bilinear numerical radii

Let A be an n-square normal matrix over $\text{C}$, and Q_{m, n} be the set of strictly increasing integer sequences of length m chosen from 1,…, n. For α,β∈Q_{m, n} denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k∈{0,1,…,m} write z.sfnc;α∩β|=k if there exists a rearrangement of 1,…,m, say i₁,…,i_k, i_k+1,…,i_m, such that α(i_j)=β(i_j), j=1,…,k, and {α(i_k+1),…,α(i_m)};∩{β(i_k+1),…,β(i_m)}=ø. Let

be the group of n-square unitary matrices. Define the nonnegative number

$\text{?}{}_{\mathrm{k}}\text{(A)=}\text{max}\text{U∈}\textcolor{red}{}\text{|}\text{det}\text{(U}{}^1\text{AU) [α|β]|}$

, where |α∩β|=k. Theorem 1 establishes a bound for ?_k(A), 0?k<m?1, in terms of a classical variational inequality due to Fermat. Let A be positive semidefinite Hermitian, n?2m. Theorem 2 leads to an interlacing inequality which, in the case n=4, m=2, resolves in the affirmative the conjecture that

$\text{?}{}_{\mathrm{m}}\text{(A)??}{}_{\mathrm{m?1}}\text{(A)????}{}_0\text{(A)}$

.     

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 $\text{(n,k)=(}\text{n?k}\text{k}\text{)+(}\text{n?k?1}\text{k?1}\text{)}$.     

Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁…,z_n), z_i ? $\text{C}$, i=1, …, n, is uniquely determined from the magnitude of f(x₁…,x_n): | f(x₁…,x_n)|, x_i ? $\text{R}$, i=1,…, n, except for (1) linear shifts: ^{i(α1z1+…+α_n2n+β}), β, α_i?$\text{R}$, i=1,…, n; and (2) conjugation: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

Extended skolem sequences

A k-extended Skolem sequence of order n is an integer sequence (s₁, s₂,…, s_2n+1) in which s_k = 0 and for each j ? {1,…,n}, there exists a unique i ? {1,…, 2n} such that s_i = s_i+j = j. We show that such a sequence exists if and only if either 1) k is odd and n ≡ 0 or 1 (mod 4) or (2) k is even and n ≡ 2 or 3 (mod 4). The same conditions are also shown to be necessary and sufficient for the existence of excess Skolem sequences. Finally, we use extended Skolem sequences to construct maximal cyclic partial triple systems. © 1995 John Wiley & Sons, Inc.     

Resolvable perfect cyclic designs

Let k, λ, and υ be positive integers. A perfect cyclic design in the class PD(υ, k, λ) consists of a pair (Q, B) where Q is a set with |Q| = υ and B is a collection of cyclically ordered k-subsets of Q such that every ordered pair of elements of Q are t apart in exactly λ of the blocks for t = 1, 2, 3,…, k?1. To clarify matters the block [a₁, a₂, …, a_k] has cyclic order a₁ < a₂ < a₃ … < a_k < a₁ and a_i and a_i+1 are said to be t apart in the block where i + t is taken mod k. In this paper we are interested only in the cases where λ = 1 and υ ≡ 1 mod k. Such a design has $\text{υ(υ ? 1)}\text{k}$ blocks. If the blocks can be partitioned into υ sets containing $\text{(υ ? 1)}\text{k}$ pairwise disjoint blocks the design is said to be resolvable, and any such partitioning of the blocks is said to be a resolution. Any set of $\text{υ ? 1)}\text{k}$ pairwise disjoint blocks together with a singleton consisting of the only element not in one of the blocks is called a parallel class. Any resolution of a design yields υ parallel classes. We denote by RPD(υ, k, 1) the class of all resolvable perfect cyclic designs with parameters υ, k, and 1. Associated with any resolvable perfect cyclic design is an orthogonal array with k + 1 columns and υ rows with an interesting conjugacy property. Also a design in the class RPD(υ, k, 1) is constructed for all sufficiently large υ with υ ≡ 1 mod k.