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Results tagged with big-list
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user 78525
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
16
votes
Important formulas in combinatorics
For a permutation $\sigma \in S_n$, let $\ell(\sigma)$ denote the maximal length of an increasing subsequence in $\sigma$. Define
$$
\ell_n = \frac{1}{n!} \sum_{\sigma \in S_n} \ell(\sigma),
$$
the av …
14
votes
Prominent non-mathematical work of mathematicians
Richard Garfield is a mathematician and former math professor who is nowadays famous as the inventor of the wildly successful card game Magic: The Gathering, and many other card and board games. Here …
11
votes
Important formulas in combinatorics
For a Young diagram $\lambda$ of size $n$, let $f^\lambda$ denote the number of standard Young tableaux of shape $\lambda$ (discussed above in Mark Wildon's answer about the hook length formula). Then …
8
votes
Special rational numbers that appear as answers to natural questions
John H. Conway's prime producing machine (also known as PRIMEGAME) is a weird algorithm that produces prime numbers using the following bizarre ordered sequence of fourteen rational numbers:
$$
\frac{ …
10
votes
Important formulas in combinatorics
I'm confused about why no one has mentioned Stirling's formula for the factorial function $n!$, clearly the most famous and important formula in asymptotic combinatorics, and easily one of the most im …
15
votes
Important formulas in combinatorics
The Rogers-Ramanujan identities are partition identities, i.e., statements that equate the number of integer partitions of an integer $n$ belonging to two different partition classes. There are two id …
25
votes
Special rational numbers that appear as answers to natural questions
I wasn't thinking of mentioning this, but some of the other answers reminded me of the elegant formulas
\begin{align}
\pi^{-6} \sum_{n,m=1}^\infty \frac{1}{(n m(n+m))^2} &= \frac{1}{2835},
\\
\pi^{-12 …
7
votes
Special rational numbers that appear as answers to natural questions
Many delicate estimates in analytic number theory lead to bounds involving unusual rational numbers. Two examples I am aware of are:
The Lindelöf hypothesis asks about the rate of growth of $|\zeta(1 …
44
votes
Special rational numbers that appear as answers to natural questions
Another example from probability theory: in critical bond percolation on the square lattice $\mathbb{Z}^2$, one can define a certain family of "connectivity events," which encode information about cer …
111
votes
32
answers
14k
views
Special rational numbers that appear as answers to natural questions
Motivation:
Many interesting irrational numbers (or numbers believed to be irrational) appear as answers to natural questions in mathematics. Famous examples are $e$, $\pi$, $\log 2$, $\zeta(3)$ etc. …
3
votes
When is 2 qualitatively different from 3?
$\mathbb{R}^d\setminus\{0\}$ is simply connected for $d\ge 3$, but not for $d=2$.
12
votes
Essays and thoughts on mathematics
The Mathematical Experience by Philip J. Davis and Reuben Hersh is a wonderful collection of essays on mathematics and on the experiences and culture of mathematicians. Written back in the 1980's, it …
13
votes
Experimental mathematics leading to major advances
The re-launching of this question is quite timely, as experimental math was behind a beautiful recent discovery of a new way to tile the plane with a pentagon. Previously only 14 such tilings were kno …
24
votes
What are some very important papers published in non-top journals?
Oded Schramm, Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000), 221--288.
This paper introduces the Schramm-Loewner Evolution (SLE), an amazing family …