Symmetric and Asymptotically Symmetric Permutations
نویسنده
چکیده
We consider two related problems arising from a question of R. Graham on quasirandom phenomena in permutation patterns. A “pattern” in a permutation σ is the order type of the restriction of σ : [n] → [n] to a subset S ⊂ [n]. First, is it possible for the pattern counts in a permutation to be exactly equal to their expected values under a uniform distribution? Attempts to address this question lead naturally to an interesting number theoretic problem: when does k! divide (n k ) ? Second, if the tensor product of a permutation with large random permutations is random-like in its pattern counts, what must the pattern counts of the original permutation be? A recursive formula is proved which uses a certain permutation “contraction.”
منابع مشابه
On Asymptotically Symmetric Banach Spaces
A Banach space X is asymptotically symmetric (a.s.) if for some C <∞, for all m ∈ N, for all bounded sequences (xj)j=1 ⊆ X, 1 ≤ i ≤ m, for all permutations σ of {1, . . . ,m} and all ultrafilters U1, . . . ,Um on N, lim n1,U1 . . . lim nm,Um ∥∥∥∥ m ∑ i=1 xini ∥∥∥∥ ≤ C lim nσ(1),Uσ(1) . . . lim nσ(m),Uσ(m) ∥∥∥∥ m ∑
متن کاملSharply $(n-2)$-transitive Sets of Permutations
Let $S_n$ be the symmetric group on the set $[n]={1, 2, ldots, n}$. For $gin S_n$ let $fix(g)$ denote the number of fixed points of $g$. A subset $S$ of $S_n$ is called $t$-emph{transitive} if for any two $t$-tuples $(x_1,x_2,ldots,x_t)$ and $(y_1,y_2,ldots ,y_t)$ of distinct elements of $[n]$, there exists $gin S$ such that $x_{i}^g=y_{i}$ for any $1leq ileq t$ and additionally $S$ is called e...
متن کاملRepresenting Random Permutations as the Product of Two Involutions
An involution is a permutation that is its own inverse. Given a permutation σ of [n], let Nn(σ) denote the number of ways to write σ as a product of two involutions of [n]. If we endow the symmetric groups Sn with uniform probability measures, then the random variables Nn are asymptotically lognormal. The proof is based upon the observation that, for most permutations σ, Nn(σ) can be well appro...
متن کاملSymmetric Permutations Avoiding Two Patterns ∗
Symmetric pattern-avoiding permutations are restricted permutations which are invariant under actions of certain subgroups of D4, the symmetry group of a square. We examine pattern-avoiding permutations with 180◦ rotational-symmetry. In particular, we use combinatorial techniques to enumerate symmetric permutations which avoid one pattern of length three and one pattern of length four. Our resu...
متن کاملSymmetric Distribution of Crossings and Nestings in Permutations of Type B
This note contains two results on the distribution of crossing numbers and nesting numbers in permutations of type B. More precisely, we prove a Bn-analogue of the symmetric distribution of crossings and nestings of permutations due to Corteel [Adv. Appl. Math. 38(2)(2007), 149–163] as well as the symmetric distribution of k-crossings and k-nestings of permutations due to Burrill et al. [DMTCS ...
متن کامل