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Aaron Robertson

Publications and source records attributed to Aaron Robertson.

31 records · Page 2Linked to original sources
arXiv2003

Avoiding Monochromatic Sequences With Special Gaps

For $S$ a set of positive integers, and $k$ and $r$ fixed positive integers, denote by $f(S,k;r)$ the least positive integer $n$ (if it exists) such that within every $r$-coloring of $\{1,2,...,n\}$ there must be a monochromatic sequence $\{x_{1},x_{2},...,x_{k}\}$ with $x_{i}-x_{i-1} \in S$ for $2 \leq i \leq k$. We consider the existence of $f(S,k;r)$ for various choices of $S$, as well as upper and lower bounds on this function. In particular, we show that this function exists for all $k$ if $S$ is an odd translate of the set of primes and $r=2$.

math.CO↗
arXiv2002

Refined Restricted Involutions

Define $I_n^k(α)$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $α\in S_i$, for some $i \geq 2$, and define $I_n^k(\emptyset;α)$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which contain the pattern $α\in S_i$, for some $i \geq 2$, exactly once. Let $i_n^k(α)$ be the number of elements in $I_n^k(α)$ and let $i_n^k(\emptyset;α)$ be the number of elements in $I_n^k(\emptyset;α)$. We investigate $I_n^k(α)$ and $I_n^k(\emptyset;α)$ for all $α\in S_3$. In particular, we show that $i_n^k(132)=i_n^k(213)=i_n^k(321)$, $i_n^k(231)=i_n^k(312)$, $i_n^k(\emptyset;132) =i_n^k(\emptyset;213)$, and $i_n^k(\emptyset;231)=i_n^k(\emptyset;312)$ for all $0 \leq k \leq n$.

math.CO↗
arXiv2002

Refined Restricted Permutations

Define $S_n^k(α)$ to be the set of permutations of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $α\in S_m$. Let $s_n^k(α)$ be the size of $S_n^k(α)$. We investigate $S_n^0(α)$ for all $α\in S_3$ as well as show that $s_n^k(132)=s_n^k(213)=s_n^k(321)$ and $s_n^k(231)=s_n^k(312)$ for all $0 \leq k \leq n$.

math.CO↗
arXiv2000

Permutations Restricted by Two Distinct Patterns of Length Three

Define $S_n(R;T)$ to be the number of permutations on $n$ letters which avoid all patterns in the set $R$ and contain each pattern in the multiset $T$ exactly once. In this paper we enumerate $S_n(\{α\};\{β\})$ and $S_n(\emptyset;\{α,β\})$ for all $α\neq β\in S_3$. The results for $S_n(\{α\};\{β\})$ follow from two papers by Mansour and Vainshtein.

math.CO↗
arXiv1999

On Generalized Van der Waerden Triples

Van der Waerden's classical theorem on arithmetic progressions states that for any positive integers k and r, there exists a least positive integer, w(k,r), such that any r-coloring of {1,2,...,w(k,r)} must contain a monochromatic k-term arithmetic progression {x,x+d,x+2d,...,x+(k-1)d}. We investigate the following generalization of w(3,r). For fixed positive integers a and b with a <= b, define N(a,b;r) to be the least positive integer, if it exists, such that any r-coloring of {1,2,...,N(a,b;r)} must contain a monochromatic set of the form {x,ax+d,bx+2d}. We show that N(a,b;2) exists if and only if b <> 2a, and provide upper and lower bounds for it. We then show that for a large class of pairs (a,b), N(a,b;r) does not exist for r sufficiently large. We also give a result on sets of the form {x,ax+d,ax+2d,...,ax+(k-1)d}.

math.CO↗
arXiv1999

Patterns and Fractions

We find, in the form of a continued fraction, the generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns, and show how to extend this to permutations that have exactly one (132) pattern. We find some properties of the continued fraction, which is similar to, though more general than, those that were studied by Ramanujan.

math.CO↗
arXiv1999

Difference Ramsey Numbers and Issai Numbers

We present a recursive algorithm for finding good lower bounds for the classical Ramsey numbers. Using notions from this algorithm we then give some results for generalized Schur numbers, which we call Issai numbers.

math.CO↗
arXiv1999

Permutations Containing and Avoiding 123 and 132 Patterns

We prove that the number of permutations which avoid 132-patterns and have exactly one 123-pattern equals (n-2)2^(n-3). We then give a bijection onto the set of permutations which avoid 123-patterns and have exactly one 132-pattern. Finally, we show that the number of permutations which contain exactly one 123-pattern and exactly one 132-pattern is (n-3)(n-4)2^(n-5).

math.CO↗