Concave Compositions: Structure & Analysis

• Concave compositions are ordered partitions structured with a decreasing sequence leading to a central distinguished part followed by an increasing sequence.
• Their generating functions, expressed via q-series and false theta functions, reveal deep links to modular and mock modular forms.
• Statistical analyses show that the distributions of part lengths converge to Gumbel and logistic laws, providing insights into asymptotic partition behavior.

A concave composition of a nonnegative integer $n$ is a finite sequence of positive integers that decreases toward a central part and then increases, forming an ordered partition that is "concave" with respect to the size of the parts. The central part, around which the sequence switches monotonicity, is referred to as the distinguished or central part. Research on concave compositions spans combinatorics, %%%%1%%%%-series, partition theory, asymptotics, and connections with modular and mock modular forms, with recent attention to their statistical structure and congruence properties.

1. Formal Definitions and Types

A concave composition of $n$ is a sequence

$$a_1 \geq a_2 \geq \cdots \geq a_r > \underline{c} < b_s \leq \cdots \leq b_2 \leq b_1,$$

where $a_j,b_j>0$ and $\underline{c}>0$ is the central part, such that

$$\sum_{j=1}^r a_j + \underline{c} + \sum_{j=1}^s b_j = n.$$

Variants arise depending on the precise monotonicities and allowed equalities:

• Even-Length (Type E): $$a_1 > a_2 > \cdots > a_m = b_m < b_{m-1} < \cdots < b_1$$, with $2m$ parts. The counting function is denoted $ce(n)$.
• Odd-Length, Type 1: $a_1 > \cdots > a_{m+1} < b_m < \cdots < b_1$, with $2m+1$ parts, counted by $co_1(n)$.
• Odd-Length, Type 2: As Type 1, except $a_{m+1} \leq b_m < \cdots < b_1$, counted by $co_2(n)$.

Strongly concave compositions enforce strict inequalities on both sides of the center.

Small examples:

• $n=3$: $ce(3)=2$, $co_1(3)=2$, $co_2(3)=3$ (Monks et al., 2014).
• $n=4$: Strongly concave compositions can be tabulated directly (see (Zhou, 2018)).

2. Generating Functions and $q$-Series Representations

The generating functions of concave compositions admit closed forms and are central to their study: $$\begin{aligned} CE(q) &= \sum_{n\geq0} ce(n) q^n = 1 + 2\sum_{k=1}^{\infty} (-1)^k \left(q^{k(3k-1)/2} + q^{k(3k+1)/2}\right), \ CO_1(q) &= \sum_{n\geq0} co_1(n) q^n = 1 - \sum_{k=1}^{\infty} (-1)^k \left(q^{6k^2-2k} + q^{6k^2+2k}\right), \ CO_2(q) &= \sum_{n\geq0} co_2(n) q^n = 1 + \sum_{k=1}^{\infty} (-1)^k \left(q^{6k^2-8k+3} + q^{6k^2-4k+1}\right), \end{aligned}$$ with $(q;q)_\infty=\prod_{m\geq1}(1-q^m)$ (Monks et al., 2014).

For the total number of concave compositions $v(n)$: $$V(q) = \sum_{n\geq0} v(n) q^n = \sum_{m\geq0} \frac{q^m}{(q^{m+1};q)_\infty^2} = (P(q))^2 \Theta^+(q),$$ where $P(q) = 1/(q)_\infty$ and $\Theta^+(q) = \sum_{n\geq0} (-1)^n q^{n(n+1)/2}$ is a false theta series (Banerjee et al., 7 Jan 2026).

3. Congruences, Parity, and Recurrences

Concave composition counts exhibit deep congruential properties:

• $ce(n)$ is even for all $n\geq1$ due to combinatorial involutions (mirror and double-zero pairings).
• For $a>2$, at least two residue classes modulo $a$ are attained by $ce(n)$ with $$\#\{n<X: ce(n)\equiv r \pmod a\} \gg \log_2\log_3 X$$ for some $r$ (Monks et al., 2014).
• Recurrences of pentagonal-type derive from generating function relations; for instance: $$ce(n) - ce(n-1) - ce(n-2) + ce(n-5) + \cdots = 0 \quad (n>0)$$ with arguments $\geq0$.

A refinement modulo 4 shows $\#\{n<X: ce(n)\equiv0\pmod4\} \gg \sqrt{X}$.

4. Combinatorial Structure, Involutions, and Partition Bijectivity

The relationship between concave compositions and classical partitions emerges via explicit combinatorial bijections:

• The number of concave compositions of even length $ce(n)$ equals the number of improper partitions of $n$—those partitions whose Durfee symbol fails a certain equality of multiplicities—constructed via an involution-based proof (Liu, 2011).
• Involution techniques operate on triples involving partitions into distinct parts and ordinary partitions, producing sign-cancelling pairs except for combinatorially meaningful survivors.

Concave composition generating functions are thus closely linked to those for certain classes of partitions (flushed, proper, improper), clarifying their enumerative and structural status.

5. Modularity, Mock Forms, and Asymptotics

The generating functions for concave compositions possess modular or "mock modular" features:

• $V(q)$ is the holomorphic part of a nonholomorphic Jacobi-type (mock Maass-theta) function of weight 1. Completion with a nonholomorphic kernel yields full modularity in the two-variable (Jacobi) setting (Banerjee et al., 7 Jan 2026).
• These generating functions bridge to false theta functions and classical partition GFs; the product $P(q)^2 \Theta^+(q)$ is a prototype of a "mixed false theta" function (Banerjee et al., 7 Jan 2026).
• Asymptotics for $v(n)$: $$v(n) \sim \frac{1}{2\sqrt{3}\, n^{3/4}} \exp\left(2\pi\sqrt{\frac{n}{3}}\right), \quad n\to\infty,$$ as established by Tauberian (Ingham) arguments using analytic properties of $V(q)$ at $q=1$.

6. Statistical Structure and Limit Laws

The probabilistic analysis of typical concave compositions, via Fristedt's conditioning device, gives a precise statistical picture (Dalal et al., 2016):

• Under the uniform measure conditioned on size, the sequence of parts on either side of the central part behave like independent partitions of size $\approx n/2$.
• The lengths $\ell(\lambda^-)$ and $\ell(\lambda^+)$ (number of parts) satisfy independent Gumbel asymptotics: $$\mathbb{P}_n \{ \ell(\lambda^\pm) \leq f_n(x) \} \to \exp(-e^{-x}),$$ for a suitable scaling function $f_n(x)$.
• The sum $\ell(\lambda^-)+\ell(\lambda^+)$ (half-perimeter) is asymptotically the sum of two independent Gumbels; the difference $\ell(\lambda^+)-\ell(\lambda^-)$ (tilt) converges to a logistic distribution.
• The expected total number of summands is

$$\mathbb{E}_n[\#\text{summands}] = \frac{2\sqrt{3n}}{\pi} \left(\ln\bigl(\frac{\sqrt{3n}}{\pi}\bigr) + \gamma_E\right) + 1 + o(\sqrt n).$$

• Limit shapes of random concave compositions involve two-sided curves characterized by shifted exponential equations; the random graph converges (after scaling) to a deterministic limit-shape up to random Gumbel shifts.

7. Rank Statistics and Associated Generating Functions

Analogous to Dyson's rank in partition theory, a rank statistic for concave (especially strongly concave) compositions encodes the asymmetry of the sequence relative to the center: $\mathrm{rk}(X) = s - 2k + 1$ for a composition $X=(a_1,\dots,a_s)$ with central part at $k$ (Zhou, 2018). The two-variable generating function for the enumeration of strongly concave compositions with prescribed rank is

$$G(x, q) = \sum_{n\geq0} \sum_{m\in\mathbb{Z}} V_a(m, n) x^m q^n = \frac{(-xq; q)_\infty\,(-x^{-1}q; q)_\infty}{(q; q)_\infty}.$$

The rank distribution, after suitable normalization, converges to the standard normal, and pointwise estimates exhibit local Gaussian behavior for the rescaled statistic.

Generalizations to restricted concave compositions, and overpartitions with accompanying rank-statistics, show that their generating functions are directly expressible in terms of classical and mixed mock theta functions, and exhibit mock modular transformation properties (Banerjee et al., 7 Jan 2026).

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For further technical detail, proofs, and more elaborate combinatorial interpretations, see (Monks et al., 2014, Liu, 2011, Dalal et al., 2016, Banerjee et al., 7 Jan 2026), and (Zhou, 2018). Each of these contributions addresses distinct facets: congruences and parity, partition bijections, probabilistic and asymptotic structure, modularity, and rank statistics, respectively.