Representation of numbers by cascades
Authors:
C. C. Chen and D. E. Daykin
Journal:
Proc. Amer. Math. Soc. 59 (1976), 394398
MSC:
Primary 05A17
MathSciNet review:
0414385
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A cascade is defined as a sum of binomial coefficients where . In this expression, we assume that whenever . Given a cascade and a sequence of signs (i.e. for each ), we define Also, we put In the paper, we prove that for any sequence of integers, there exist a cascade and a corresponding sequence of signs such that for where , and recursively, .
 [1]
Joseph
B. Kruskal, The number of simplices in a complex, Mathematical
optimization techniques, Univ. of California Press, Berkeley, Calif.,
1963, pp. 251–278. MR 0154827
(27 #4771)
 [2]
G.
Katona, A theorem of finite sets, Theory of graphs (Proc.
Colloq., Tihany, 1966) Academic Press, New York, 1968,
pp. 187–207. MR 0290982
(45 #76)
 [3]
D.
E. Daykin, Jean
Godfrey, and A.
J. W. Hilton, Existence theorems for Sperner families, J.
Combinatorial Theory Ser. A 17 (1974), 245–251. MR 0416931
(54 #4993)
 [4]
D.
E. Daykin, A simple proof of the KruskalKatona theorem, J.
Combinatorial Theory Ser. A 17 (1974), 252–253. MR 0416932
(54 #4994)
 [5]
, Cascade algorithms giving Katonatype inequalities, Nanta. Math. (to appear).
 [6]
, The average size set in an antichain, Nanta. Math. (to appear).
 [1]
 J. B. Krexuskal, The number of simplices in a complex, Mathematical Optimization Techniques, Univ. of California Press, Berkeley, 1963, pp. 251278. MR 27 #4771. MR 0154827 (27:4771)
 [2]
 G. Katona, A theorem of finite sets, Theory of Graphs (Proc. Colloq., Tihany, 1966), Academic Press, New York, 1968, pp. 187207. MR 45 #76. MR 0290982 (45:76)
 [3]
 D. E. Daykin, Jean Godfrey and A. J. W. Hilton, Existence theorems for Sperner families, J. Combinatorial Theory Ser. A 17 (1974), 245251. MR 0416931 (54:4993)
 [4]
 D. E. Daykin, A simple proof of the KruskalKatona theorem, J. Combinatorial Theory Ser. A 17 (1974), 252253. MR 0416932 (54:4994)
 [5]
 , Cascade algorithms giving Katonatype inequalities, Nanta. Math. (to appear).
 [6]
 , The average size set in an antichain, Nanta. Math. (to appear).
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
05A17
Retrieve articles in all journals
with MSC:
05A17
Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919760414385X
PII:
S 00029939(1976)0414385X
Article copyright:
© Copyright 1976 American Mathematical Society
