Ira Gessel
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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.