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This is inspired by Fedor's answer: Consider $$ f(m):=-\frac{\binom{ab}{m}\binom{cd}{m}(m-bc)(m-ad)}{\binom{ad-1}{m}\binom{bc-1}{m}(a-c)(b-d)}. $$ Then $$ f(m+1)-f(m)=-\frac{\binom{ab}{m+1}\binom …

This is a bit of cheating (since requires to know a bit of homological algebra) but is too nice to not be mentioned: Let's notice that the generating function for dimensions of graded components fo …

The LHS of your formula can be rewritten as $$ \sum_{k\ge 1, d\mid k}\left[\frac{n}{k}\right]\mu(k/d)X^d, $$ which after rearranging terms becomes $$ \sum_{d\ge 1}X^d\sum_{s\ge 1}\mu(s)\left[\frac{ …

Another formula (almost without alternating signs) can be obtained as a variation of the comment of David Speyer. Namely, for each $S\subset\{1,\ldots,k\}$ we can consider the set of all monomials tha …

It just crossed my mind that there is another way to compute the cardinality of the complement (and I decided to post it as well to demonstrate the power of generating functions): it is the coefficien …

Surely there are many: these are all polynomials in one variable, so every two of them are algebraically dependent because of the transcendence degree argument :-) However, I am sure that this is not …

Now, to the matter of "positive formulas". Classical result of Kraskiewicz and Weyman (preprint W. Kraskiewicz, J. Weyman. Algebra of Invariants and the Action of a Coxeter Element. Math. Inst. Copern …

For your question about Lie triple systems: In my article "Veronese powers of operads and pure homotopy algebras" joint with Martin Markl and Elisabeth Remm, Eur. J. Math. 6 (2020), 829-863 (https://l …

(This is a bit too long for a comment, though not exhaustive at all.) This number, especially if you make appropriate changes of your notation to replace $< $ by $\le$ (replace $a_i$ by $a_i-i$, and …

For rather trivial reasons, $$ 1+\sum_{(k,l)\ne (0,0)}p(k,l)x^ky^l=\prod_{(p,q)\ne(0,0)}\frac{1}{1-x^py^q} . $$ Since these numbers include, as $p(n,0)$, the one-dimensional partition numbers, you …

Composition of species is closely related to the composition of symmetric collections of vector spaces ("S-modules"), which is a remarkable example of a monoidal category everyone who had ever encount …

Gjergji Zaimi already said it all, but I want to point out a tiny bit longer but equally cute way to derive the same formula. Every monic polynomial over $\mathbb{F}\_q$ decomposes into a product of i …

The following is too long for a comment, so let me type it as an answer though it does not literally answer your question. Using the standard formula $$ T\_{2k-1}=(-1)^{k-1}2^{2k}(2^{2k}-1)\frac{B_{ …

What kind of a formula will you find satisfactory? Formulas for the plethysm $s_\lambda\circ h_n$ where coefficients are expressed in terms of $S_n$-characters and generalized Kostka numbers are in Ma …

The answer (already given in comments, with a small misprint/mistake) is: (a) for even $r=2k+2$, it is $q^{m-1}+(q-1)q^{m-k-2}\eta((-1)^{k+1})$, where $\eta$ is the quadratic character of $\mathbb{F} …