All Questions
84
questions
-1
votes
0
answers
10
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Column/Digit blind solution for the "Number of possible combinations of x numbers that sum to y"
What formula will give me "the total number of possible combinations of x digits that sum to y".
This is a branch question from the original question entitled
Number of possible ...
5
votes
1
answer
63
views
Recursion regarding number-partitions
I am learning about partitions of numbers at the moment.
Definition:
Let $n \in \mathbb{N}$. A $k$-partition of $n$ is a representation of $n$ as the sum of $k$ numbers greater than $0$, (i.e.
$n=a_1+....
- 609
1
vote
1
answer
38
views
Equality between q shifted factorials and sum with partitions
I am looking at WilliamY.C.Chen,Qing-HuHou,and Alain Lascoux proof for the Gauss Identity in q shifted factorials. At some point they claim that it is easy to see that
$$
\frac{q^r}{(q ; q)_r}=\sum_{\...
2
votes
2
answers
174
views
Number of ways to complete a partial Young tableau
Suppose we have a Young tableau with missing entries. Then there can be many number of ways we can complete the Young Tableau.
Is there any specific method to find the number of ways we can complete a ...
- 399
2
votes
1
answer
78
views
the bracket partition function ? $3 = 1 + 1 + 1 = 1 + 2 = 1 + (1 + 1) $
I want to know in how many ways we can write a positive integer by using strict positive integers, addition and brackets. The order of addition does not matter.
For instance
$$3 = 1 + 1 + 1 = 1 + 2 = ...
- 16.4k
1
vote
1
answer
86
views
Number of partitions with distinct even parts/parts with multiplicity $\leq 3$
I am supposed to solve a problem regarding partitions of $n \in \mathbb{N}$ into:
distinct even parts
parts with multiplicity $\leq 3$
I am supposed to prove that 1. and 2. are equal.
So I tried ...
- 333
0
votes
0
answers
75
views
Question about number of generating functions
I know that the generating function for the number of integer partitions of $n$ into distinct parts is
$$\sum_{n \ge 0} p_d(n)x^n = \prod_{i \ge 1}(1+x^i)$$
I'm trying to use this generating function ...
- 27
1
vote
1
answer
185
views
Find the number of non-negative integer solutions to $x+y+z=11$
How do I find the number of nonnegative integer solutions to $x+y+z=11$ provided that $x\leq 3, y\leq 4, z\leq 6$ using the sum rule (counting)?
I know the answer is 6, but I'm having difficulty ...
- 2,072
3
votes
1
answer
109
views
Distributing $n$ distinct objects into $m$ types of urns with $k_1,k_2...k_m$ urns of each type
I came accross this (rather complex?) combinatorial problem:
I have $18$ distinct objects, $3$ red urns, $7$ blue urns, and $11$
green urns. In how many ways can I distribute the objects into those
...
- 1,022
4
votes
1
answer
255
views
Alternative way of writing the stars and bars formula where each bar is associated with at least one star.
I was looking for a different way of writing the formula of the number of different $k$-tuples of non-negative integers whose sum is equal to $n$ and I thought of this formula followed by this ...
- 51
1
vote
0
answers
62
views
Number of solutions to linear equation $x_1+x_2+\dots+x_n=m$ when the domain of $x_i\ne$ domain of $x_j$
In the lecture notes of one of my previous classes, it was used that if we have an equation of the form
$$\tag{1}
x_1+x_2+\dots+x_n=m
$$
then the total number of solutions, when each $x_i$ is a non-...
- 175
0
votes
1
answer
113
views
Counting number of possible sequences given two constraints
Consider a 6 sided dice which takes values from $\{1,2,...,6\}$. let $n_i$ denote the number of times $i \in [6]$ appears on the dice. From this dice we create a non-decreasing sequence $(a_1,...,a_6)$...
- 392
0
votes
1
answer
95
views
Prove combinatorially the recurrence $p_n(k) = p_n(k−n) + p_{n−1}(k−1)$ for all $0<n≤k$.
Recall that $p_n(k)$ counts the number of partitions of $k$ into exactly $n$ positive parts (or, alternatively, into any number of parts the largest of which has size $n$).
- 31
1
vote
1
answer
78
views
Identity with integer partitions
I have to prove that $p(n)=p(n-1)+\displaystyle\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}p_{k}(n-k)$ and I am quite stuck on it...
My first intuition was that, as $p(n)-p(n-1)$ is the number of partitions ...
- 763
1
vote
1
answer
172
views
Is a Standard Tableau determined by its descent set?
Suppose $\lambda\vdash n$ is a partition. Associated with this partition is the set of Standard Young Tableau $\text{SYT}(\lambda)$ such that the associated Young Diagram is filled in with the numbers ...
- 405