Black Cell Capacity

• Black cell capacity is a discrete statistic that quantifies black cells via a parity-based coloring in bargraph representations of combinatorial constructs, including words and polyominoes.
• Enumerative frameworks employ generating functions and closed-form formulas to precisely count black cells in k-ary words, permutations, and Catalan polyominoes.
• In energy storage studies, black cell capacity helps model lithium-ion cell performance through adaptive diagnostics and efficient capacity estimation.

Black cell capacity is a discrete statistic defined via a parity-based coloring of the cells in bargraph representations of combinatorial or physical constructs, most prominently words over finite alphabets, permutations, and structurally constrained polyominoes. The notion quantifies the count of those cells colored black under a specified chessboard convention, and has been the focus of recent enumerative and probabilistic investigations in combinatorics as well as an adapted term in modeling capacity-related features in energy storage devices.

1. Definition and General Construction

The black cell capacity for a bargraph is rooted in tiling a 2D grid: each column indexed by $i$ has integer height $h_i$, and each cell at coordinate $(i, j)$ (with $1 \leq j \leq h_i$) is colored black if $i + j$ is even, white otherwise. This chessboard-style coloring ensures that adjacent cells differ in color along any path orthogonal to either axis.

Given a $k$-ary word $u = u_1u_2\cdots u_n$, the associated bargraph consists of $n$ columns, the $i$-th of height $u_i$. For a permutation $\pi \in S_n$, one uses the sequence of values as heights. The black and white cell statistics are given by

$\black(u) = \sum_{i=1}^n \black_i(u_i), \qquad \white(u) = \sum_{i=1}^n \white_i(u_i)$

where

$\black_i(h) = \begin{cases} \lceil h/2 \rceil, & \text{%%%%11%%%% odd} \ \lfloor h/2 \rfloor, & \text{%%%%12%%%% even} \end{cases} \qquad \white_i(h) = \begin{cases} \lfloor h/2 \rfloor, & \text{%%%%13%%%% odd} \ \lceil h/2 \rceil, & \text{%%%%14%%%% even} \end{cases}$

Thus, the black cell capacity is the total $\black(u)$ for $u$, and, in the context of Catalan polyominoes, the same rules apply to the diagram induced by the respective height sequence (Fried, 9 Sep 2025, Baril et al., 24 Jan 2026).

2. Enumerative Framework: Generating Functions and Formulas

$k$-ary words:

The bivariate generating function recording black and white cell counts is

$f_n(b, w) = \sum_{u \in [k]^n} b^{\black(u)} w^{\white(u)}, \qquad F_k(x; b, w) = \sum_{n \ge 0} f_n(b, w) x^n$

with

$$g_k(b, w) = \sum_{h=1}^k b^{\lceil h/2 \rceil}w^{\lfloor h/2 \rfloor} = \frac{b(1-(bw)^{\lceil k/2 \rceil}) + bw(1-(bw)^{\lfloor k/2 \rfloor})}{1 - bw}$$

then

$$F_k(x ; b, w) = \frac{1 + x g_k(b, w)}{1 - x^2 g_k(b, w) g_k(w, b)}$$

The independence per position and parity leads to multiplicative structural factors in the enumeration (Fried, 9 Sep 2025).

Permutations:

For $\pi\in S_n$,

$f_n(b, w) = \sum_{\pi \in S_n} b^{\black(\pi)} w^{\white(\pi)}$

admits a closed form in terms of Jacobi polynomials $P_m^{(\alpha,0)}$: $$f_n(b, w) = (bw)^{\lfloor n^2/4 \rfloor} \lfloor n/2 \rfloor! \lceil n/2 \rceil! b^\alpha (w-b)^{\lfloor n/2 \rfloor} P_{\lfloor n/2 \rfloor}^{(\alpha, 0)}\left(\frac{w+b}{w-b}\right)$$ with $$\alpha = \lceil n/2 \rceil - \lfloor n/2 \rfloor \in \{0,1\}$$, derived via permanents of weight matrices and summation identities (Fried, 9 Sep 2025).

Catalan Polyominoes:

For polyomino $P$ associated to Catalan word $w_1\cdots w_n$, the black cell capacity $B(P)$ is recorded in the bivariate series

$$G(x, q) = \sum_{n\geq 1} \sum_{P \in \mathbf{C}_n} x^{n} q^{B(P)}$$

A 4-component parity-refined system expresses $G(x,q)$, with a recursive matrix formulation and, via a bijection, a “vertical capacity” formula reducing the counting to explicit Pochhammer sums. This closed form captures the distribution of the black cell statistic over all Catalan polyominoes (Baril et al., 24 Jan 2026).

3. Balanced Words, Exact Counts, and Asymptotics

A bw-balanced word is defined by $\black(u) = \white(u)$. The key enumerative result is

$$\mathrm{bal}_k(n) = \sum_{r=0}^{\lfloor n/2 \rfloor} \binom{\lceil n/2 \rceil}{r} \binom{\lfloor n/2 \rfloor}{r} \lceil k/2 \rceil^{2r} \lfloor k/2 \rfloor^{n-2r}$$

If $k$ is even, this further simplifies to

$$\mathrm{bal}_k(n) = \left(\frac{k}{2}\right)^{n} \binom{n}{\lfloor n/2 \rfloor}$$

The generating function is algebraic: $$\mathrm{BAL}_k(x) = \sum_{n \geq 0} \mathrm{bal}_k(n) x^n = \frac{1}{\Delta_k} \left[ 1 + \lfloor k/2 \rfloor x + \frac{1-(\lfloor k/2 \rfloor^2 + \lceil k/2 \rceil^2)x^2 - \Delta_k}{2 \lfloor k/2 \rfloor x} \right]$$ with

$$\Delta_k = \sqrt{(1 - (\lfloor k/2 \rfloor^2 + \lceil k/2 \rceil^2)x^2)^2 - 4\lfloor k/2 \rfloor^2 \lceil k/2 \rceil^2 x^4}$$

Asymptotically, for even $k$, the fraction of balanced words among all $k$-ary words of length $n$ is $\sim \sqrt{2/(\pi n)}$; for permutations, positive asymptotic density arises only on lengths $n \equiv 0,3 \pmod{4}$, with limiting proportion $\sim \sqrt{8/(\pi n)}$ (Fried, 9 Sep 2025).

4. Catalan Polyominoes: Structure and Distribution

A length-$n$ Catalan word $w_1w_2\cdots w_n$ (with $w_1=0$, $w_i \leq w_{i-1} + 1$ for all $i$) encodes a column-convex polyomino, with the chessboard coloring applied as above. The black cell capacity $B(P)$ is tracked combinatorially via specialized bivariate generating functions, with functional equations partitioned by the parity of the width and last column. A bijective correspondence demonstrates that the distribution of $B(P)$ coincides with the vertical black (or white) capacity across the set. Matrix continued fractions and infinite series produce an explicit formula in terms of $q$-Pochhammer symbols (Baril et al., 24 Jan 2026).

Enumerative data illustrate the distribution. For example, for polyominoes of width 4, there are 2 objects with $B=2$, 5 with $B=3$, 4 with $B=4$, 2 with $B=5$, and 1 with $B=6$. The sequence of total counts (over all widths and capacities) does not align to existing OEIS entries.

5. Practical Capacity Concepts in Lithium-Ion Cells

In physical sciences, “black cell capacity” functions as a descriptive or convenient term for gravimetric or volumetric energy capacity in Li-ion cells, particularly in studies involving full cells with conversion anodes and high-voltage spinel cathodes. For an $\alpha$-Fe$_2$O$_3$@C / LiNi$_{0.5}$Mn$_{1.5}$O$_4$ cell, key capacity metrics are as follows:

• $$Q_{\mathrm{cell, cathode}} = 110\,\mathrm{mAh\,g}^{-1}$$ (referred to cathode active mass)
• $$Q_{\mathrm{cell, total}} \approx 83\,\mathrm{mAh\,g}^{-1}$$ (active materials only)
• Average discharge voltage $V_{\mathrm{avg}} \approx 3.2\,\mathrm{V}$
• Specific energy $E_{\mathrm{cathode}} = 352\,\mathrm{Wh\,kg}^{-1}$, $$E_{\mathrm{total}} \approx 266\,\mathrm{Wh\,kg}^{-1}$$

After accounting for inactive mass (e.g., SEI, binder, current collector), a practical whole-cell energy density of $\sim$150 Wh kg$^{-1}$ is reported. The system achieves a Coulombic efficiency $>97\%$ over 60 cycles and a capacity retention of $\sim$96\% (115 $\rightarrow$ 110 mAh g$^{-1}$ at C/2). Performance is competitive with LiFePO$_4$ systems and other CuO/spinel full cells, with the key technical advantage of cobalt-free, low-cost electrode design (Wei et al., 2021).

6. Estimation of Cell Capacity in Model-Based Battery Diagnostics

Real-time estimation of usable (cyclable) Ah capacity in Li-ion batteries is formalized in model-based observers, notably via the Enhanced Single Particle Model (ESPM). Here, the total lithium inventory in the negative electrode, $Q(t)$, is modeled as a slowly-varying state, explicitly tracking loss mechanisms (e.g., SEI layer growth) with

$\frac{dQ}{dt} = -\frac{i_s a_{s,n} A L_n}{3600}$

An adaptive, Lyapunov-stable sliding-mode observer updates $Q$ online, estimating both electrode transport parameters and capacity under physical, temperature-dependent conditions. Onori and Allam achieve bounded error in $Q$ estimation (error $<2\%$ across different states of health), robust even under sensor bias and stochastic noise (Allam et al., 2020). This approach operationalizes “black box” or “black cell capacity” as a trackable model state under experimentally relevant conditions.

7. Illustrative Examples and Edge Cases

• For the $k$-ary word $u = 1\,5\,2\,3\,2\,2$, the bargraph yields $\black(u) = 7$, $\white(u) = 8$ (Fried, 9 Sep 2025).
• For length-3 permutations, the distribution is:
  • $123,\;321 \mapsto b^4 w^2$
  • $132,\;213,\;231,\;312 \mapsto b^3 w^3$

Thus, $f_3(b,w) = 2b^4w^2 + 4b^3w^3$.

• For $k=2$, $n=2$, the balanced words are “11” and “22”; the formula $\mathrm{bal}_2(2) = (1)^2 \cdot 2 = 2$ is exact.

Tabular summary for $k$-ary words and permutations:

Structure	Statistic	Formula / Generating Function
$k$-ary words	$f_n(b,w)$	$$F_k(x; b, w) = \frac{1 + xg_k(b, w)}{1 - x^2g_k(b, w)g_k(w, b)}$$
Permutations	$f_n(b,w)$	$$f_n(b,w)=\dots P_{\lfloor n/2 \rfloor}^{(\alpha, 0)}\left(\frac{w+b}{w-b}\right)$$
Catalan polyomino	$G(x, q)$	Explicit functional/matrix and Pochhammer sum forms

These analytical constructs serve both enumerative and applied modeling purposes and provide a foundation for future investigations in parity-coloring statistics for more general combinatorial and applied system classes.