Let $p(n)$ denote the number of partitions of a positive integer $n$. It is known that $\{p(n)\}_{n>25}$ is log-concave. Dietrich Burde said in this MathOF post that property $PF_3$ for partition function $p(n)$ semms to hold for all $n>221$. And he said that the question is also motivated from the study of Betti numbers for nilpotent Lie algebras, in particular filiform nilpotent Lie algebras.

But I try to find the relationship between Partition function and Betti numbers, but failed.

Need Help.

  • $\begingroup$ If you're satisfied by this answer, could you mark it as answered? $\endgroup$ – YCor Jun 18 '17 at 16:43

The relationship is as follows. Denote by $\mathfrak{f}_n$ the standard graded filiform nilpotent Lie algebra of dimension $n$. Then the following result is known.

Theorem (Armstrong, Sigg 1996): Let $L$ be a complex nilpotent Lie algebra having an abelian ideal of codimension $1$. Then the sequence of Betti numbers $(b_i(L))_{i\ge 1}$ is unimodal. In particular, the sequence $(b_i(\mathfrak{f}_n))_{i\ge 1}$ is unimodal.

There are formulas for the Betti numbers of $\mathfrak{f}_n$ in terms of restricted partitions. The $r$-th Betti number of $\mathfrak{f}_{n+1}$ is given by $$ b_r(\mathfrak{f}_{n+1})=P_{r,n}+P_{r-1,n}$$ for $1\le r \le n+1$, where $P_{0,n}=1$ and $$P_{r,n}=\# \Bigl \{ (a_1,\ldots ,a_r)\in \mathbb{Z}^r \mid 1\le a_1 < \dots < a_r \le n,\quad \sum_{j=1}^r a_j =\Bigl \lceil \frac{r(n+1)}{2} \Bigr \rceil \Bigr \}.$$ An explicit formula is only known for small $r$: \begin{align*} b_1(\mathfrak{f}_{n}) & = 2 \\ b_2(\mathfrak{f}_{n}) & =\left \lfloor \frac{n+1}{2} \right \rfloor ,\\ b_3(\mathfrak{f}_{n}) & = \left \lfloor \binom{\frac{n+1}{2}}{2}+\frac{1}{8} \right \rfloor = \left \lfloor \frac{n^2}{8}\right \rfloor,\\ b_4(\mathfrak{f}_{n}) & = \left \lfloor \frac{4}{3}\binom{\frac{n+1}{2}}{3}+ \frac{4n+13}{36} \right \rfloor = \left \lfloor \frac{(n-1)^3+18}{36}\right \rfloor .\\ \end{align*}

Computing these Betti numbers $b_i$ for $n\le 50$ I found that they form a log-concave sequence, for $i\ge 2$. So the the following question came up:

Question: Is it true that the sequence of Betti numbers $(b_i(\mathfrak{f}_n))_{i\ge i_0}$ for some $i_0>1$ is log-concave?

The first thing which one would like to use for a proof here is that the ordinary partition function itself is log-concave. Already this was not clear to me. Hence my question at MO.


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