5
votes
Accepted
Permutation graph with insert-and-shift
Diameter equals $n-1$.
For an upper bound, we should get from every permutation an identical one by at most $n-1$ such transformations. Just place 1 to its place (one transformation is always enough), ...
5
votes
Number of regions created by r hyper-planes in n-dimensional space
One proof is in my book Enumerative Combinatorics, vol. 1, second ed., Proposition 3.11.8. The first proof is due to L. Schläfli, written in 1850-52 but not published until 1901 in Neue allgemeinen ...
4
votes
Order on Euclidean space in which a finite poset embeds
If $(P,\le)$ is a poset, the least $n$ such that $(P,\le)$ embeds into a product of $n$ total orders (or equivalently, such that $\le$ is the intersection of $n$ total orders on $P$) is known as the ...
4
votes
A question related to "Locally Sidorenko" type problem
If I understand your question correctly, you are asking if there is a sparse graph counting lemma assuming only that the graphon is within $o(p)$ to the density $p$ in cut norm; the Lovasz result ...
4
votes
What is this Ramsey problem?
A good reference is Radziszowski's article Small Ramsey Numbers, which gets updated as new results are proven. In particular, this refers to basically all known Ramsey style results. The ones you're ...
3
votes
Accepted
Ask for a generating function or an explicit expression of a triangle of positive integers
The generating function:
$${\cal C}(x,y) = \sum_{n,k\geq 0} C_{n,k} x^n y^{2k}$$
has the following explicit form:
$${\cal C}(x,y) = \frac{\arctan(y)}{y(1-x(1+y^2))}.$$
For "one more problem",...
3
votes
How to efficiently sample uniformly from the set of $p$-partitions of an $n$-set?
I have provided some Python code for a quick and dirty implementation of the algorithm described by Kevin in the hope that it will save someone some time. The randomPartitionsFixedP() function is the ...
1
vote
Accepted
Formula for partitions of integers with no subpartition being a partition of $t$
Let $t$ be fixed.
Per Answer 1, the number of 2-forcing (nonnegative) partitions equals the coefficient of $q^M$ in Gaussian binomial coefficient $\binom{N+t-1}{N}_q$.
To answer Question 1.5, it is ...
1
vote
Homotopical Combinatorics
Back when this question was asked, several people suggested they were not sure what was meant by "homotopical combinatorics." Well, now there's a subfield known as homotopical combinatorics. ...
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