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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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