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A couple of survey papers are here and here. There is also a book by Mara Neusel and Larry Smith.

Write $E_j=\sum_{i=0}^j \frac{x^i}{i!}$, and let $F_m(b_1,\dots,b_k)$ denote the coefficient of $x^m/m!$ in the product $E_{b_1}\cdots E_{b_k}$. By standard properties of exponential generating functi …

Curtis Greene et al. have a nice package at http://ww3.haverford.edu/math/cgreene.html.

Philippe Flajolet and Robert Sedgewick's Analytic Combinatorics is the most comprehensive reference, available at http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html. Also useful is Odly …

J.S. Provan and M.O. Ball, The complexity of counting cuts and of computing the probability that a graph is connected, SIAM J. Comput. 12 (1983) 777-78, show that the problem of computing the number o …

I can answer my own question. The reference is I. J. Schoenberg, Regular simplices and quadratic forms, J. London Math. Soc. 12 (1937), 48-55.

The second identity is the special case $x=1$, $z=1$, $y=0$ of Abel's generalization of the binomial theorem, as stated in Enumerative Combinatorics, vol. 2, Exercise 5.31(c). This identity states tha …

The group algebra over $\mathbb{R}$ of every finite group is a direct sum of simple algebras. If you decide to compute this decomposition for some small groups, then you will discover that for the 8-e …

This is a routine application of the exponential formula. If $S$ is any subset of the positive integers and $f_S(n)$ is the number of partitions of $[n]$ into parts all belonging to $S$, then $$ \su …

To the contrary, it can be shown that any two SYT of the same shape are related by a sequence of flips. An explicit negative answer is given by $$ \begin{array}{ll}1246 & 1246\\ 35 & 37\\ 7 & 5 \end …

Let $k$ be the dimension over $\mathbb{Q}$ of the $\mathbb{Q}$-subspace of $\mathbb{R}$ spanned by $a_1,\dots,a_d$. Then the dimension over $\mathbb{Q}$ of all $d$-tuples $(n_1,\dots,n_d)\in\mathbb{Q} …

For shifted RSK, see Section 3.5 of the thesis of Luis Guillermo Serrano Herrera at http://deepblue.lib.umich.edu/bitstream/handle/2027.42/77864/lserrano_1.pdf;jsessionid=B5A8F5AD166B6960A30EE5E16394A …

There is a Java applet that displays all 5-element connected posets at http://www1.chapman.edu/~jipsen/gap/posets.html.

The subset order on intervals of a finite poset has received some attention. See for example Exercises 3.10, 3.76(b), 3.138, and 3.158(c) of http://math.mit.edu/~rstan/ec/ec1.pdf.

The correct generating function is $$ \exp\left( x +\frac{x^2}{2}+\frac 12\sum_{n\geq 2}\frac{x^{2n}}{2n}\right) =\frac{\exp\left(x+\frac{x^2}{4}\right)}{(1-x^2)^{1/4}}. $$ This appears in http …