52 research outputs found
Strongly intersecting integer partitions
We call a sum a1+a2+• • •+ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ • • • ≤ ak and n = a1 + a2 + • • • + ak. For i = 1, 2, . . . , k, we call ai the ith part of the sum a1 + a2 + • • • + ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1+a2+• • •+ak and b1+b2+• • •+bk strongly intersect if ai = bi for some i. We call a subset A of Pn,k strongly intersecting if every two partitions in A strongly intersect. Let Pn,k(1) be the set of all partitions in Pn,k whose first part is 1. We prove that if 2 ≤ k ≤ n, then Pn,k(1) is a largest strongly intersecting subset of Pn,k, and uniquely so if and only if k ≥ 4 or k = 3 ≤ n ̸∈ {6, 7, 8} or k = 2 ≤ n ≤ 3.peer-reviewe
On the Spectrum of the Derangement Graph
We derive several interesting formulae for the eigenvalues of the derangement graph and use them to settle affirmatively a conjecture of Ku regarding the least eigenvalue
Complete intersection theorem and complete nontrivial-intersection theorem for systems of set partitions
We prove the complete intersection theorem and complete
nontrivial-intersection theorem for systems of set partitionsComment: add aknowlegments and distribute material between two my papers in
arxiv in another orde
Cross-intersecting families and primitivity of symmetric systems
Let be a finite set and , the power set of ,
satisfying three conditions: (a) is an ideal in , that is,
if and , then ; (b) For with , if for any
with ; (c) for every . The
pair is called a symmetric system if there is a group
transitively acting on and preserving the ideal . A
family is said to be a
cross--family of if for any and with . We prove that if is a
symmetric system and is a
cross--family of , then where . This generalizes Hilton's theorem on
cross-intersecting families of finite sets, and provides analogs for
cross--intersecting families of finite sets, finite vector spaces and
permutations, etc.
Moreover, the primitivity of symmetric systems is introduced to characterize
the optimal families.Comment: 15 page
Cross-intersecting families of permutations
For positive integers r and n with r n, let Pr,n be the family of all sets {(1, y1), (2, y2),. . . , (r, yr)} such that y1, y2,..., yr are distinct elements of [n]={1, 2,...,n}. Pn,n describes permutations of [n]. For r < n, Pr,n describes permutations of r-element subsets of [n]. Families A1,A2,...,Ak of sets are said to be cross-intersecting if, for any distinct i and j in [k], any set in Ai intersects any set in Aj. For any r, n and k 2, we determine the cases in which the sum of sizes of cross-intersecting sub-families A1,A2,...,Ak of Pr,n is a maximum, hence solving a recent conjecture.peer-reviewe
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