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 Yearling
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Feb
3
revised Elementary+Short+Useful
corrected spelling and grammar
Jan
29
comment Existence of analytic continuation of $f(z)=\sum{n^{\alpha}} z^n$ for fractional $\alpha$
This is the polylogarithm; see en.wikipedia.org/wiki/Polylogarithm, especially the section on integral representations.
Jan
23
comment Alternative definition of the Lagrange Inversion formula
A combinatorial approach to Lie series has been given by Gilbert Labelle in two papers: MR0814421 (87c:05007) Une combinatoire sous-jacente au théorème des fonctions implicites. [A combinatorial theory underlying the implicit function theorem] J. Combin. Theory Ser. A 40 (1985), no. 2, 377–393 and MR0787718 (86j:05015) Éclosions combinatoires appliquées à l'inversion multidimensionnelle des séries formelles. [Combinatorial bloomings applied to the multidimensional inversion of formal series] J. Combin. Theory Ser. A 39 (1985), no. 1, 52–82.
Jan
7
comment Alternative definition of the Lagrange Inversion formula
This kind of expansion is called a Lie series. I didn't check the author's derivation, but formulas of this type are well known. Unfortunately I don't know of a really good introductory reference.
Dec
30
revised Combinatorial meaning of the functional equation for logarithm
deleted 2 characters in body
Dec
27
comment Asymptotic growth rate of coefficients of generating function
You might also look at F. Harary, R. W. Robinson, and A. J. Schwenk, Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc. 20 (Series A) (1975), 483-503. link
Dec
4
comment Why does $d^n \exp(-x-x^{-1})/(dx)^n$ only have $n$ positive real zeroes?
Also by Taylor's theorem, the polynomials $\phi_n(y)$ have the generating function $$\sum_{n=0}^\infty \phi_n(y) \frac{z^n}{n!} = \exp\left(-z\left(1-\frac{y^2}{1+yz}\right)\right). $$
Nov
23
comment what is the link between plethysm in regular representation of the symmetric group and plethysm in Schur functions.
Here is an explanation of $s_2\circ s_2$ in ‘simple terms‘. Here we have permutation representations: $s_2$ is the characteristic of $S_2$ acting on the single set $\{1,2\}$ by permutating 1 and 2. This is the trivial representation, since $S_2$ is acting on a single element. The plethysm $s_2\circ s_2$ is the characteristic of $S_4$ acting on partitions of $\{1,2,3,4\}$ into two blocks of size 2; i.e. acting on the three partitions $\{\{1,2\},\{3,4\}\},$ $\{\{1,3\},\{2,4\}\},$ and $\{\{1,4\},\{2,3\}\},$ by permuting 1, 2, 3, and 4.
Nov
23
comment Extraction of Coefficients in the Exponential Function of a Series
Note that $g=e^f$ implies $g' = gf'$. This gives a recurrence for the coefficients of $g$ that's easy to use.
Nov
21
comment A remarkable sum over partitions
Alternatively, equate coefficients of $x^n$ in $$\frac{1}{1-x} = \exp\biggl(\sum_{a=1}^\infty \frac{x^a}{a}\biggr).$$
Nov
11
awarded  Yearling
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.