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Jul
13 |
revised |
Generate all non-isomorphic partitions $\pi = \{ \{1, …, n-1\}, \{n\} \}$ for all graphs of order $n$
added 5 characters in body |
Jul
3 |
revised |
Computer package for representation theory of the symmetric group
fixed a mistake in (1) |
Jun
23 |
answered | Generate all non-isomorphic partitions $\pi = \{ \{1, …, n-1\}, \{n\} \}$ for all graphs of order $n$ |
Jun
18 |
awarded | Civic Duty |
Jun
13 |
comment |
A new generalisation of Fermat's little theorem?
This result has also been attributed to Ramachandra (mathoverflow.net/questions/87048/…). |
Jun
12 |
comment |
A new generalisation of Fermat's little theorem?
According to Dickson's History of the Theory of Numbers, Volume 1, p. 84 (archive.org/details/historyoftheoryo01dick) this generalization of Fermat's theorem is due to Gauss, at least when $a$ is a prime. On p. 82 Dickson gives a reference for the general case to Thue in 1910, though it may be older. |
Apr
27 |
comment |
Counting distinct undirected, partially labelled graphs
There's a proof of Pólya's theorem by Gian-Carlo Rota and David Smith using Möbius inversion along the lines that gowers suggests, but the usual proof of Burnside's lemma is much easier. See Rota, Gian-Carlo; Smith, David A. Enumeration under group action. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Sér. 4, 4 no. 4 (1977), p. 637-646 numdam.org/item?id=ASNSP_1977_4_4_4_637_0 |
Apr
11 |
comment |
convergence radius of Pochhammer symbol series
This is not a power series, so it doesn't have a radius of convergence. |
Apr
11 |
revised |
convergence radius of Pochhammer symbol series
Improved formatting and punctuation. |
Feb
20 |
answered | Number of Dyck paths with prescribed number of edges |
Feb
13 |
comment |
A combinatorial identity generalizing identity (3.111) from Gould's book
If we change the upper limit of the sum to $n$ then the sum is the $n$th difference of a polynomial of degree $n-1$ and is therefore 0. The only nonzero term in the extended sum not in the restricted sum is $-1$, corresponding to $m=n$. |
Feb
1 |
comment |
Identity for Power Series and Binomial Coefficients
(2) can be derived from (1) by subtracting 1, dividing by $r$ and taking the limit as $r\to 0$. |
Jan
26 |
answered | Identity for Power Series and Binomial Coefficients |
Jan
20 |
comment |
Probability question involving drawing balls from an urn
One way to do this is with the transfer matrix method. (See, e.g., chapter 4 of Richard Stanley's Enumerative Combinatorics, volume 1.) The basic idea is that as you draw the balls, you keep track of the colors of the last two balls drawn. You can represent the possible transitions as edges in a directed graph with weights that keep track of the number of R's, B's, and triples of each type, so that the numbers you want can be obtained by extracting coefficients from powers of a 4 by 4 matrix, and from this matrix you can get a rational generating function for the numbers. |
Jan
11 |
comment |
Binomial coefficient identity
This identity is also a special case of Vandermonde's theorem. |
Dec
26 |
answered | A recurrence relation on Catalan numbers |
Dec
8 |
comment |
Counting path generating sentences in a specific formal language
For an arbitrary Turing machine there is little or nothing that can be said. |
Dec
7 |
comment |
Nontrivial question about Fibonacci numbers?
More generally, for $p\ne 2$ or 5, $F_{n+p} \equiv F_{n+1} \pmod p$ if $(p|5)=1$ and $F_{n+p}\equiv -F_{n-1} \pmod p$ if $(p|5) = -1$. (Here $F_{-1}=1$.) |
Dec
5 |
comment |
A natural sum over multisets (expectation over multinomial)
The generating function $f(x)=\sum_n n^n x^n/n!$ is equal to $1/(1+W(-x))$, where $W$ is the Lambert $W$-function. The denominator has a zero at $x=1/e$ which should enable you to get a good approximation to the coefficients of $f(x)^k$ by standard techniques (see, e.g., Flajolet and Sedgewick's Analytic Combinatorics). |
Nov
30 |
revised |
Dyck paths on rectangles
improved TeX formatting |