How many way can we write the integer n as the sum of non nigative integer such that the largest number of the k non-negative integers is d. The condition is the largest number in all k non-negative must be dd. For example, I can write 10 as 3+3+2+1+1, 3+3+2+1+1 and 3+2+2+2+1, 3+2+2+2+1 if k=5 and d=3.
This is a very classical subject and it is called the number of partitions of a given number.
have a look at https://en.wikipedia.org/wiki/Partition_(number_theory)
Partitions with a bound on the highest length (the leg length) are important in the classification of young diagrams (which in turn are important in the representation of groups) and are classified by so called Jack polynomials.
https://en.wikipedia.org/wiki/Jack_function
This is a very classical subject and it is called the number of partitions of a given number.
have a look at https://en.wikipedia.org/wiki/Partition_(number_theory)
Partitions with a bound on the highest length (the leg length) are important in the classification of young diagrams (which in turn are important in the representation of groups) and are classified by so called Jack polynomials.
https://en.wikipedia.org/wiki/Jack_function
The number #(n) of ways to write a positive integer n as the sum of natural numbers are given as follows:
#(2)=2, #(3)=3, #(4)=5, #(5)=7, #(6)=11, #(7)=15, #(8)=22, #(9)=30, #(10)= 42, #(11)=55, #(12)=77, ..., #(20)=627, ..., #(50)=204226,..., #{100)=190569291,..., #(200)=3972999029388, ...
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10th Oct, 2016
Frank Janssens
Classical References:
Hardy&Wright : Introduction to the theory of numbers
Vinogradov: Elements of number theory
I don't know if the constraints that you add to the partition problem have been treated
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1 Recommendation
10th Oct, 2016
Loring Tu
Tufts University
In Bang-Yen Chen's answer, #(7) should be 15, not 14. By dropping the zero terms, the number of ways #(n) of writing a positive integer n as a sum nonnegative integers is the same as the number of ways of writing n as a sum of positive integers. So #(n) is exactly the well-known partition function p(n) in classical number theory. There is a large literature on p(n), for example, in Wikipedia.
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13th Oct, 2016
Vemula Ramakrishna Reddy
VIT University AP
This is nothing but the partition of a natural number. I am agreeing with Dr Rogier Brussee. Please look at the link
https://en.wikipedia.org/wiki/Partition_(number_theory)
and you can find this topic on many books like Abstract Algebra by Herestein, Contemporary Abstract Algebra by Joseph A Gallian and Abstract Algebra by Richard M. Foote David S. Dummit.
I hope this help you
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14th Oct, 2016
Eman F. Mohommed
Al-Mustansiriya University
please i have two condition in my composition of n
1- length = k
2- large number must be = d
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15th Oct, 2016
Yaser Maleki
Tarbiat Modares University
this question was return to Ramanujan and before than Ramanujan and i think this is unsolvable until know
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