Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Leapfrog in Posets   We consider the following solitaire game whose rules are reminiscent of the
children's game of leapfrog. The player is handed an arbitrary ordering
$\pi=(x_1,x_2,...,x_n)$ of the elements of a finite poset $(P,\prec)$. At each
round an element may "skip over" the element in front of it, i.e. swap
positions with it. For example, if $x_i \prec x_{i+1}$, then it is allowed to
move from $\pi$ to the ordering
$(x_1,x_2,...,x_{i-1},x_{i+1},x_i,x_{i+2},...,x_n)$. The player is to carry out
such steps as long as such swaps are possible. When there are several
consecutive pairs of elements that satisfy this condition, the player can
choose which pair to swap next. Does the order of swaps matter for the final
ordering or is it uniquely determined by the initial ordering? The reader may
guess correctly that the latter proposition is correct. What may be more
surprising, perhaps, is that this question is not trivial. The proof works by
constructing an appropriate system of invariants.

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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: patt-sol
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The categories of this paper are: adap-org
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas