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Differential posets and Smith normal forms. (English) Zbl 1228.05096
Summary: We conjecture a strong property for the up and down maps $$U$$ and $$D$$ in an $$r$$-differential poset: $$DU + tI$$ and $$UD + tI$$ have Smith normal forms over $${\mathbb Z}[t]$$. In particular, this would determine the integral structure of the maps $$U, D, UD, DU$$, including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice $$\mathbf Y F$$ studied by Okada and its $$r$$-differential generalizations $$Z(r)$$, as well as verifying many of its consequences for Young’s lattice $$\mathbf Y$$ and the $$r$$-differential Cartesian products $$Y ^{r }$$.

##### MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 05A15 Exact enumeration problems, generating functions 05A17 Combinatorial aspects of partitions of integers 06A07 Combinatorics of partially ordered sets
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