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{\bf R. Milson}
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\noindent {\bf Composition sum identities related to the
distribution of coordinate values in a discrete simplex. }
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Utilizing spectral residues of parameterized, recursively defined
sequences, we develop a general method for generating identities of
composition sums. Specific results are obtained by focusing on
coefficient sequences of solutions of first and second order,
ordinary, linear differential equations.
Regarding the first class, the corresponding identities amount to a
proof of the exponential formula of labelled counting. The
identities in the second class can be used to establish certain
geometric properties of the simplex of bounded, ordered, integer
tuples.
We present three theorems that support the conclusion that the inner
dimensions of such an order simplex are, in a certain sense, more
ample than the outer dimensions. As well, we give an algebraic
proof of a bijection between two families of subsets in the order
simplex, and inquire as to the possibility of establishing this
bijection by combinatorial, rather than by algebraic methods.
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