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Sequences by Number of w-Rises

Published online by Cambridge University Press:  20 November 2018

Morton Abramson
Morton Abramson*
Affiliation:
York University, Toronto, Ontario, Canada
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An m-permutation of n, repetitions allowed, is an m-sequence

(1)

A w-rise is a pair (ei, ei+1) such that ei+1-ei≥w>0. In this note we find an expression for Tk, w(n, m), the number of m-sequences having precisely k w-rises. The case w = 1 is given in [1] [2]. Also, when w = 1 we give the number when each of the integers 1, 2, …, r must appear at least once.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 3 , August 1975 , pp. 317 - 319
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Carlitz, L., Roselle, D. P. and Scoville, R. A., Permutations and sequences with repetitions by number of increases, J. Comb. Theory 1 (1966), pp. 350-374.Google Scholar
2. Dillon, J. F. and Roselle, D. P., Simon Newcomb's problem, SIAM J. Appl. Math. 17 (1969), pp. 1086-1093.Google Scholar
3. Gould, H. W., Combinatorial Identities, West Virginia University, Morgantown, West Virginia, 1972.Google Scholar
4. Moser, W. O. J. and Morton, Abramson, Enumeration of Combinations with restricted differences andcospan, J. Comb. Theory 7 (1969), pp. 162-170.Google Scholar
5. Riordan, J., An introduction to combinatorial analysis, J. Wiley, New York, 1958.Google Scholar