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Results tagged with co.combinatorics
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user 40145
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
26
votes
6
answers
2k
views
For which $n$ is there a permutation such that the sum of any two adjacent elements is a prime?
For which $n$ is it possible to find a permutation of the numbers from $1$ to $n$ such that the sum of any two adjacent elements of the permutation is a prime?
For example: For $n=4$ the permutation …
17
votes
1
answer
795
views
Are There Always Group Generators Which Give Unimodal Growth?
Suppose $G$ is a $k$-generated finite group. Is there always a set of $k$ elements which generate the group and have a unimodal counting function?
Background:
The counting function, $f(n)$, is a fun …
16
votes
0
answers
454
views
A Product Related to Unrestricted Partitions
Start with the product for unrestricted partitions:
$(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...$
Now replace some of the plus signs with minus signs and expand the product into a series. Is it …
13
votes
3
answers
1k
views
Can you make an identity from this product?
Start with the product
$$(1+x+x^2) (1+x^2)(1+x^3)(1+x^4)\cdots$$
(The first polynomial is a trinomial..The others are binomials..)
Is it possible by changing some of the signs to get a series all of …
9
votes
1
answer
187
views
A simplified version of an old problem about the generating function for unrestricted partit...
This is my latest attempt to simplify an old problem of mine so much that the simplified problem can actually be answered.
Starting with the generating function for unrestricted partitions:
$$(1+x+x …
8
votes
0
answers
200
views
A Product Related to Partitions with Largest Part n
This is a finite version of a problem of mine entitled "A product related to unrestricted partitions." It has the advantage that, at least for small values of n, it is easily solved.
Begin with the …
7
votes
1
answer
217
views
Partitions restricted to only certain summands
Here $n$ is a positive integer and $p(n)$ is the number of unrestricted partitions.
Can one always find a subset $s$, of $\{1,2,\ldots,n\}$ such that the number of partitions of $n$ with parts fro …
7
votes
1
answer
218
views
Another question related to the generating function for unrestricted partitions
This question is similar to another that I asked, but should be, I think, very much easier.
Start with the generating function for unrestricted partitions and replace some of the plus signs with minu …
6
votes
1
answer
299
views
an identity related to the pentagonal numbers
How can I prove the following?
$$1-x+x^2+x^5-x^7-x^{12}+x^{15}-x^{22}-x^{26}+x^{35}-x^{40}+\dots \\= \prod_{i=1}^{\infty} [(1 - x^{8 i - 7}) (1 + x^{8 i - 6}) (1 + x^{8 i - 5}) (1 +
x^{8 i - 4} …
4
votes
1
answer
293
views
A sequence reminiscent of Fibonacci's recursion
The sequence in question is A296768 in the Online Encyclopedia. It starts with
1, 3, 5, 9, 11, 17, 24, 32, 36, 46, ...
It is obtained by starting with the positive integers in order, (b(i)= i for all …
4
votes
0
answers
146
views
Another Generalization of a Problem of Steinhaus
In his book of problems from elementary mathematics Hugo Steinhaus asked the following:
Is it possible to find an infinite sequence of real numbers $x_1,x_2,...$, such that
$x_1$ lies in the interva …
2
votes
1
answer
470
views
A Problem Related to the 17 Point Problem of Steinhaus
In his book of problems from elementary mathematics Hugo Steinhaus asked the following:
does there exist for every positive integer N a sequence of real numbers $x_1,x_2,...,x_N$ such that for every …
1
vote
0
answers
205
views
Can you make these two series equal?
Consider the two series defined by
series 1: $$(1+x+x^2+x^3+...) (1+x^3+x^6+x^9...)(1+x^5+x^{10}+x^{15}+...)...$$
and
series 2: $$1+x+x^2+x^5+x^7+...$$ Where the exponents 1,2,5,7,...are the pentagon …
1
vote
0
answers
113
views
A question about partitions into distinct parts
This question is a variant of the question posed by Brian Hopkins.
Let $\operatorname{pdist}(n)$ be the number of partitions of $n$ into distinct parts. Also, let $\operatorname{pdist}(S, n)$ denote …
1
vote
1
answer
224
views
Will this greedy algorithm always work?
Let p(n) be the number of unrestricted partitions of n. p(0) is taken to be 1.
Let set 1 and set 2 be two empty sets.
Here's an algorithm. Put p(n) into set 1. On each successive step, k=1,2,3,..., …