Nonisomorphic Ordered Sets with Arbitrarily Many Ranks That Produce Equal Decks

Details Nonisomorphic Ordered Sets with Arbitrarily Many Ranks That Produce Equal Decks Nonisomorphic Ordered Sets with Arbitrarily Many Ranks That Produce Equal Decks

2006-09-25

arxiv.org/abs/math/0609702v1

Mathematics

Combinatorics

30 pages, 6 figures, straight LaTeX

Scientific paper

We prove that for any $n$ there is a pair $(P_1 ^n , P_2 ^n )$ of nonisomorphic ordered sets such that $P_1 ^n $ and $P_2 ^n $ have equal maximal and minimal decks, equal neighborhood decks, and there are $n+1$ ranks $k_0 , \ldots , k_n $ such that for each $i$ the decks obtained by removing the points of rank $k_i $ are equal. The ranks $k_1 , \ldots , k_n $ do not contain extremal elements and at each of the other ranks there are elements whose removal will produce isomorphic cards. Moreover, we show that such sets can be constructed such that only for ranks $1$ and $2$, both without extremal elements, the decks obtained by removing the points of rank $r_i $ are not equal.

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