8
votes
0answers
411 views

Reference/quote request: “All of combinatorics is the representation theory of $S_n$”

I think I remember reading somewhere a glib (or is it deep?) quote, perhaps due to Rota?, which was something like "All of combinatorics is essentially [or can be reduced to?] the representation ...
3
votes
0answers
121 views

Citation for subset complement result

Let $S=\lbrace s_1,\ldots,s_n \rbrace \subset \lbrace1,\ldots,2n\rbrace$. Consider two operations on $S$: the complement $C(S)=\lbrace 1,\ldots,2n \rbrace \setminus S$ and a reflection* ...
6
votes
3answers
650 views

Classification of Platonic solids

My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula ...
5
votes
1answer
221 views

constant averages along orbits

What should one say to describe the situation in which a function $T$ from some set $X$ to itself, and a function $f$ from $X$ to some characteristic-zero field $K$, have the property that the average ...
3
votes
1answer
184 views

Historical question about MacMahon theorem, Wronski relation etc…

Seemingly the same fact goes under several names: MacMahon master theorem, Wronski relation, unnamed fact about symmetric functions - I wonder what is history and what should be ``correct name'' ? ...
13
votes
0answers
840 views

Origins of the Nerve Theorem

Recently, I've read two papers which have cited the Nerve Theorem, one crediting Borsuk with the result and another Leray. Here is the question: Who was the first to prove the Nerve Theorem?
3
votes
1answer
408 views

Klein Bottle exception to the Heawood Conjecture [duplicate]

Possible Duplicate: The Klein bottle and the Heawood Conjecture It is well known that the Heawood Conjecture states that the bound for the number of colours which are sufficient to colour a ...
7
votes
3answers
778 views

What is so “plactic” about the plactic monoid?

The plactic monoid is the monoid consisting of all words from the alphabet $\mathbb{Z}^+$ modulo certain relations. It is important mainly because its elements enumerate semistandard Young tableaux. ...
5
votes
1answer
448 views

Where does the definition of “Tower of Algebras” come from?

A tower of algebras is a sequence of algebras $$A_0 \hookrightarrow A_1 \hookrightarrow \cdots \hookrightarrow A_n \hookrightarrow \cdots$$ with embeddings $A_n \otimes A_m \hookrightarrow A_{n+m}$ ...
19
votes
1answer
1k views

Fibonacci, compositions, history

There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets): a) compositions with parts from {1,2} (e.g., 2+2 = 2+1+1 = ...
4
votes
3answers
958 views

Why were plane partitions invented?

I realize that these objects were originally created by Major Percy Macmahon and today have many applications but what was the original motivation for studying them?
7
votes
9answers
430 views

What are some early examples of creation of lists / catalogues of (particularly) combinatorial objects?

A lot of effort in discrete maths / combinatorics is expended in the construction of lists, catalogues or census [sic] of combinatorial objects such as groups, graphs, designs etc. These catalogues ...
3
votes
2answers
755 views

Birkhoff's theorem about doubly stochastic matrices

Birkhoff's theorem states: The set of $n \times n$ doubly stochastic matrices is a convex set whose extreme points are the permutation matrices This theorem seems to be commonly attributed to ...
24
votes
3answers
1k views

Does there exist a comprehensive compilation of Erdos's open problems?

Fan Chung and Ron Graham's book Erdos on Graphs: His Legacy of Unsolved Problems (A. K. Peters, 1998) collects together all of Erdos's open problems in graph theory that they could find into a single ...
29
votes
4answers
6k views

Do actual Sudoku puzzles have a unique rational solution?

Here is a question in the intersection of mathematics and sociology. There is a standard way to encode a Sudoku puzzle as an integer programming problem. The problem has a 0-1-valued variable ...
17
votes
4answers
2k views

Genealogy of the Lagrange inversion theorem

A wonderful piece of classic mathematics, well-known especially to combinatorialists and to complex analysis people, and that, in my opinion, deserves more popularity even in elementary mathematics, ...
7
votes
5answers
1k views

Texts on the General History of Contemporary Combinatorics

I am looking for some core texts (books, book chapters, papers) about the general history of contemporary combinatorics, starting, say, from the interwar period up to today. Texts about the history ...