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Note on a combinatorial problem

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Abstract

In the first part of this paper we have assembled some properties of the quantitiesR n m , whereR n m denotes the number of distributions ofn different objects intom indifferent parcels, with no empty parcels allowed. We then discuss the following problem (N. Rashevsky, 1954, 1955 a,b, 1956): to find the total number,G n , of graphs that can be obtained from the biotopological transformation (T (1) X) for a given value of the parametern. This is related to the distribution ofn indifferent objects intom different boxes. A formula forG n is given which, however, is not very convenient for practical computations because it involves a summation over certain “admissible partitions” of the numbermn (m is a second parameter of the transformation). Some theorems are derived; with their help we can simplify the calculation ofG n to a small extent. The numbersG n are calculated forn≤9 and estimated forn=10. It is found thatG 7≈5.4×104,G 8≈8.3×105,G 9≈1.4×107, andG 10≈3×108. These values ofn are those which might be used in connection with N. Rashevsky’s work (cf. Rashevsky, 1956).

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Authors and Affiliations

  1. Mental Health Research Institute, University of Michigan, USA

    Ernesto Trucco

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  1. Ernesto Trucco
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Trucco, E. Note on a combinatorial problem. Bulletin of Mathematical Biophysics 19, 309–336 (1957). https://doi.org/10.1007/BF02478420

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  • DOI: https://doi.org/10.1007/BF02478420

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