5,153 research outputs found
Generalized hyperbolic functions, circulant matrices and functional equations
There is a contrast between the two sets of functional equations f_0(x+y) =
f_0(x)f_0(y) + f_1(x)f_1(y), f_1(x+y) = f_1(x)f_0(y) + f_0(x)f_1(y), and
f_0(x-y) = f_0(x)f_0(y) - f_1(x)f_1(y), f_1(x-y) = f_1(x)f_0(y) - f_0(x)f_1(y)
satisfied by the even and odd components of a solution of f(x+y) = f(x) f(y).
J. Schwaiger and, later, W. F\"org-Rob and J. Schwaiger considered the
extension of these ideas to the case where f is sum of n components. Here we
shorten and simplify the statements and proofs of some of these results by a
more systematic use of matrix notation.Comment: 18 pages; corrected and updated versio
The Rees product of posets
We determine how the flag f-vector of any graded poset changes under the Rees
product with the chain, and more generally, any t-ary tree. As a corollary, the
M\"obius function of the Rees product of any graded poset with the chain, and
more generally, the t-ary tree, is exactly the same as the Rees product of its
dual with the chain, respectively, t-ary chain. We then study enumerative and
homological properties of the Rees product of the cubical lattice with the
chain. We give a bijective proof that the M\"obius function of this poset can
be expressed as n times a signed derangement number. From this we derive a new
bijective proof of Jonsson's result that the M\"obius function of the Rees
product of the Boolean algebra with the chain is given by a derangement number.
Using poset homology techniques we find an explicit basis for the reduced
homology and determine a representation for the reduced homology of the order
complex of the Rees product of the cubical lattice with the chain over the
symmetric group.Comment: 21 pages, 1 figur
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