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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 …
David S. Newman's user avatar
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 …
David S. Newman's user avatar
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 …
David S. Newman's user avatar
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 …
David S. Newman's user avatar
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 …
David S. Newman's user avatar
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 …
David S. Newman's user avatar
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 …
David S. Newman's user avatar
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 …
David S. Newman's user avatar
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} …
David S. Newman's user avatar
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 …
David S. Newman's user avatar
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 …
David S. Newman's user avatar
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 …
David S. Newman's user avatar
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 …
David S. Newman's user avatar
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 …
David S. Newman's user avatar
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,..., …
David S. Newman's user avatar

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