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Restricted Partitions of Finite Sets

Published online by Cambridge University Press:  20 November 2018

F.L. Miksa ,
L. Moser  and
M. Wyman
M. Wyman
Affiliation:
University of Alberta
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In this paper we consider the following combinatorial problem. In how many ways can n distinguishable objects be placed into an unrestricted number of indistinguishable boxes, if each box can hold at most r objects? Let us denote this number by Gn, r

Special cases of this problem have been the object of considerable study. In the case r = 2 we have the numbers Gn, 2 = Tn which have been treated by Rothe [12] as early as 1800. Tn is also the number of solutions of x2 = 1 in the symmetric group on n letters , and in this and related guises has been studied by Touchard [13], Chowla, Herstein and Moore [3] and two of the present authors [7].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 1 , Issue 2 , May 1958 , pp. 87 - 96
Copyright
Copyright © Canadian Mathematical Society 1958

References

1. Becker, H. W., On the general theory of rhyme (abstract), Ball. Amer. Math. Soc. 52 (1946), 415.Google Scholar
2. Cernuschi, F. and Castegnette, L., Chains of rare events, Annals of Math. Statist. 17 (1946), 53-61.Google Scholar
3. Chowla, S., Herstein, I. N. and Moore, K., On recursions connected with symmetric groups I, Can. J. Math. 3 (1951), 328-334.Google Scholar
4. Finlayson, H., Numbers generated by eex-1, Master's thesis, University of Alberta (1954).Google Scholar
5. Gupta, H., Tables of distributions, Res. Bull. East Punjab Univ. (1950), 13-44.Google Scholar
6. Hadwiger, H., Gruppierung mit Nebenbedingungen, Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker. 43 (1943), 113-123.Google Scholar
7. Moser, L. and Wyman, M., On solutions of xd = 1 in symmetric groups, Can. J. Math. 7 (1955), 159-168.Google Scholar
8. Moser, L. and Wyman, M., On an array of Aitken, Proc. Roy. Soc. Can. (Sec. III) 48 (1954), 31-37.Google Scholar
9. Moser, L. and Wyman, M., Asymptotic formula for the Bell numbers, Proc. Roy. Soc. Can. (Sec. III) 49 (1955), 49-54.Google Scholar
10. Moser, L. and Wyman, M., Asymptotic expansions. Can. J. Math. 8 (1956), 225-233.Google Scholar
11. Netto, E., Lehrbuch der Combinatorik (zweite Auflage, Leipzig 1927), Nachtrag von Viggo Brun, 283-286.Google Scholar
12. Rothe, H.A., Űber Permutationen in Beziehung auf die Stelle ihrer Elemente, Sammlung combinatorischer analytischer Abhandlungen, herausgegeben v. C.F. Hindenburg (zweite Sammlung, Leipzig 1800), 263-305.Google Scholar
13. Touchard, J., Sur les cycles des substitutions, Acta Math, 70 (1939), 243-297.Google Scholar