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Snevily, Hunter S.


Combinatorics

Roller Coaster permutations ★★★

Author(s): Ahmed; Snevily

Let $ S_n $ denote the set of all permutations of $ [n]=\set{1,2,\ldots,n} $. Let $ i(\pi) $ and $ d(\pi) $ denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in $ \pi $. Let $ X(\pi) $ denote the set of subsequences of $ \pi $ with length at least three. Let $ t(\pi) $ denote $ \sum_{\tau\in X(\pi)}(i(\tau)+d(\tau)) $.

A permutation $ \pi\in S_n $ is called a Roller Coaster permutation if $ t(\pi)=\max_{\tau\in S_n}t(\tau) $. Let $ RC(n) $ be the set of all Roller Coaster permutations in $ S_n $.

Conjecture   For $ n\geq 3 $,
    \item If $ n=2k $, then $ |RC(n)|=4 $. \item If $ n=2k+1 $, then $ |RC(n)|=2^j $ with $ j\leq k+1 $.
Conjecture  (Odd Sum conjecture)   Given $ \pi\in RC(n) $,
    \item If $ n=2k+1 $, then $ \pi_j+\pi_{n-j+1} $ is odd for $ 1\leq j\leq k $. \item If $ n=2k $, then $ \pi_j + \pi_{n-j+1} = 2k+1 $ for all $ 1\leq j\leq k $.

Keywords:

Posted by Tanbir Ahmed
updated October 14th, 2013
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Number Theory » Combinatorial N.T.

Snevily's conjecture ★★★

Author(s): Snevily

Conjecture   Let $ G $ be an abelian group of odd order and let $ A,B \subseteq G $ satisfy $ |A| = |B| = k $. Then the elements of $ A $ and $ B $ may be ordered $ A = \{a_1,\ldots,a_k\} $ and $ B = \{b_1,\ldots,b_k\} $ so that the sums $ a_1+b_1, a_2+b_2 \ldots, a_k + b_k $ are pairwise distinct.

Keywords: addition table; latin square; transversal

Posted by mdevos
updated May 27th, 2011
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