--- title: 'Sqrank: Partition Statistic & LoRA Scaling' url: https://www.emergentmind.com/topics/sqrank type: topic --- # Sqrank: Partition Statistic & LoRA Scaling Searching arXiv for papers relevant to “Sqrank,” including partition-theoretic and LoRA-related usages. I’m going to look up the most relevant arXiv entries for “sqrank” and closely related usages. Sqrank is a polysemous technical term in current arXiv literature. In partition theory, it denotes a recently introduced statistic on integer partitions defined through the Durfee square, repeated rim-hook removal, and Frobenius coordinates; its distribution is linked to odd minimal excludants, Gaussian $q$-binomials, configuration sums, and affine-crystal combinatorics [2412.19503]. In parameter-efficient fine-tuning for large language models, “Sqrank” is also used for the rank-stabilized LoRA scaling rule that replaces the original $\alpha/r$ factor by $\alpha/\sqrt r$ in low-rank adapters [2312.03732]. These usages are distinct, and both should also be separated from “stable rank” in representation-geometry-based alignment and from classical rank queries in succinct data structures [2512.02807; 0907.1103]. ## 1. Partition-theoretic definition For an integer partition $\lambda\vdash n$ with Ferrers diagram $\lambda$, the partition-theoretic sqrank is defined from the Durfee square $D_0(\lambda)$ of side $$ n_0(\lambda)=d_0. $$ Let $A_0(\lambda)$ be the subdiagram consisting of the first $n_0$ rows of $\lambda$ outside $D_0(\lambda)$. From $A_0(\lambda)$, one repeatedly removes the longest of the rightmost rim hooks of arm-length $n_0$ until the residual diagram $R_0(\lambda)$ has fewer than $n_0+1$ columns. If the Frobenius representation of $R_0(\lambda)$ is $$ F\bigl(R_0(\lambda)\bigr)=(x_1,\dots,x_d\mid y_1,\dots,y_d), $$ with $$ y_0=n_0,\qquad x_{d+1}=-1, $$ then $$ \mathrm{sqrank}(\lambda)=\max_{0\le i\le d}(y_i-x_{i+1})-1. $$ The same source states an equivalent interpretation: sqrank measures how “deep” one can strip horizontal–vertical rim-hooks of length $n_0$ from the subdiagram to the right of the Durfee square, and then reads off the maximal difference of leg–arm lengths in the residual shape plus an offset [2412.19503]. A later affine-crystal treatment restates the same statistic as $$ \mathrm{sqrank}(\lambda)=f^{(0)}(\lambda)=\max_{0\le i\le d}\bigl(y_i-x_{i+1}\bigr)-1, $$ and adds an equivalent characterization in terms of a $0$–$1$ encoding of $R_0(\lambda)$. If $R_0(\lambda)$ is encoded as a binary string $\eta$ of length $2n_0$ via $$ \eta=(1^{y_d},\,0^{x_d+1},\;1^{y_{d-1}-y_d},\,0^{x_{d-1}-x_d},\;\dots,\;1^{n_0-y_1},0^{n_0-x_1-1}), $$ then sqrank is the unique $r$ with $0\le r\le n_0$ such that the Kashiwara function satisfies $\varepsilon_1(\eta)=r$ [2605.30806]. ## 2. Enumerative formulas and the odd minimal excludant Fix $r\ge 0$, and define $$ p^{(r)}_{2,1}(n)=\#\{\lambda\vdash n\mid \mathrm{sqrank}(\lambda)=r\}. $$ The generating function given for this distribution is $$ \sum_{n\ge0}p^{(r)}_{2,1}(n)\,q^n =\sum_{m\ge0}\frac{q^{m^2}\,Z_{2m,m}^{(r)}(q)}{(q;q)_{2m}}, $$ where $$ Z_{2m,m}^{(r)}(q)=\genfrac{[}{]}{0pt}{}{2m}{\,m-r}-\genfrac{[}{]}{0pt}{}{2m}{\,m-r-1}. $$ The same generating function is also written in closed form as $$ \sum_n p^{(r)}_{2,1}(n)\,q^n=\frac{q^{r^2}(1-q^{2r+1})}{(q;q)_\infty}. $$ This leads to the principal equidistribution statement: if $\mathrm{mex}_{2,1}(\lambda)$ is the smallest positive odd integer not appearing as a part of $\lambda$, then $$ p^{(r)}_{2,1}(n)=\#\{\lambda\vdash n\mid \mathrm{mex}_{2,1}(\lambda)=2r+1\} =\#\{\lambda\vdash n\mid \mathrm{sqrank}(\lambda)=r\}. $$ The paper presents this as Theorem 2.1(1) and proves it by deriving the same generating function from mex-count arguments and from an energy-preserving bijection through restricted partitions and bit-sequences [2412.19503]. The crystal-theoretic treatment gives the same one-variable generating function in the notation $$ p_{sq}(n,r)=\#\{\lambda\vdash n:\mathrm{sqrank}(\lambda)=r\}, $$ namely $$ \sum_{n\ge0} p_{sq}(n,r)\,q^n =\frac{q^{r^2}(1-q^{2r+1})}{(q;q)_\infty}, $$ and also records the two-variable series $$ \sum_{\lambda\in\mathcal P} q^{|\lambda|}z^{\mathrm{sqrank}(\lambda)} =\sum_{r\ge0}\frac{q^{r^2}(1-q^{2r+1})}{(q;q)_\infty}z^r. $$ From the identical generating function for odd mex, it concludes that for each $n,r$, partitions of $n$ with sqrank $r$ are equinumerous with those of odd mex $2r+1$, and in particular that $0\le \mathrm{sqrank}(\lambda)\le n_0(\lambda)$ [2605.30806]. ## 3. Bosonic polynomials, bit-paths, and affine crystals A central structural feature of sqrank is its appearance in a polynomial bosonic form. The polynomial $$ Z_{L,s}(q)=\sum_{\eta}q^{E(\eta)} $$ enumerates length-$L$ bit sequences with $s$ ones by the energy $$ E(\eta)=\sum_{j=1}^{L-1}j\,H(\eta_j,\eta_{j+1}),\qquad H(0,1)=1\ \text{else }0. $$ The unrefined configuration sum satisfies $$ Z_{L,s}(q)={L\brack s}, $$ while the refined form $$ Z_{L,s}^{(r)}(q)={L\brack s-r}-{L\brack s-r-1} $$ counts paths of fixed path-minimum $r$. The same source states that these exactly match the statistical-mechanics configuration sums of an integrable box–ball automaton, and that sqrank arises via an energy-preserving mapping from partitions to bit-paths [2412.19503]. The same work isolates the special case $$ X^+_{L,s}(q)=Z_{L,s}^{(0)}(q), $$ with recursion $$ X^+_{L,s}=X^+_{L-1,s}+\sum_{k=1}^s q^{L-k}\,X^+_{L-k-1,s-k}, $$ boundary conditions $$ X^+_{L,0}=1,\qquad X^+_{2s-1,s}=0, $$ and solution $$ X^+_{L,s}(q)={L\brack s}-{L\brack s-1}. $$ This places sqrank within a family of Gaussian-polynomial identities rather than as an isolated partition statistic [2412.19503]. In affine-crystal language, sqrank controls the position of a partition inside $U_q(\widehat{\mathfrak{sl}_2})$ representation-theoretic structure. For $B(\Lambda_0)$, the level-$1$ crystal, there is an explicit bijection $$ \lambda\vdash n,\ \mathrm{sqrank}(\lambda)=r \ \longmapsto\ p(\lambda;\Lambda_0)\in B(\Lambda_0) $$ such that the image path is $f_1$-highest, lies in the $(2r+1)$-node $\mathfrak{sl}_2$-component, and has left-energy $$ E_{\leftarrow}^{\Lambda_0}(p)=n. $$ The construction proceeds by encoding $R_0(\lambda)$ as a binary string, applying $e_1^r$ to obtain an $f_1$-highest element in a tensor crystal, inflating adjacent $01\to0011$, and then inserting blocks of type $10$ or $01$ according to the remaining partition data. The resulting bijection preserves grading [2605.30806]. The 2026 paper also interprets the same structure in the Bernard–Pasquier–Serban spinon picture. For a vacuum-sector path, if the $i$-th string carries two spinons of momenta $$ k_{2i}=u_i-i,\qquad k_{2i-1}=v_i-i, $$ then $$ E_{\leftarrow}^{\Lambda_0}(p)=\sum_{i=1}^N (u_i+v_i-1)=\sum_{j=1}^{2N}k_j+N^2. $$ It then states that sqrank enters as $r=N$ in the $(2r+1)$-dimensional component and controls the total number of spinon excitations and the form of the Virasoro generator $L_0$ spectrum in the spinon basis [2605.30806]. ## 4. Worked examples and source-level discrepancies The supplied literature contains incompatible worked examples for partitions of $4$. One source lists the five partitions of $4$ and assigns the following sqrank values: $\lambda=1^4\mapsto 0$, $\lambda=2\,1^2\mapsto 1$, $\lambda=2^2\mapsto 2$, $\lambda=3\,1\mapsto 0$, and $\lambda=4\mapsto 2$ [2412.19503]. A later source gives instead: $\mathrm{sqrank}(4)=0$, $\mathrm{sqrank}(3,1)=1$, $\mathrm{sqrank}(2,2)=2$, $\mathrm{sqrank}(2,1,1)=0$, and $\mathrm{sqrank}(1,1,1,1)=1$ [2605.30806]. | Partition of $4$ | Value in [2412.19503] | Value in [2605.30806] | |---|---:|---:| | $(4)$ | $2$ | $0$ | | $(3,1)$ | $0$ | $1$ | | $(2,2)$ | $2$ | $2$ | | $(2,1,1)$ | $1$ | $0$ | | $(1,1,1,1)$ | $0$ | $1$ | The 2026 source explicitly flags a correction issue for $\lambda=(4)$ by first writing a computation that would suggest a different value and then stating: “Careful,” followed by the assertion that the explicit bijection in Section 1 yields $\mathrm{sqrank}(4)=0$ [2605.30806]. A plausible implication is that small-$n$ examples require consultation of the full combinatorial convention being used, especially for the empty residual diagram $R_0(\lambda)$ and its interaction with the crystal-theoretic normalization. ## 5. “Sqrank” in rank-stabilized LoRA In large-language-model fine-tuning, “Sqrank” is used as a name for the rank-stabilized LoRA scaling factor. For a pre-trained linear layer $$ y=W\,x+b,\qquad W\in\mathbb R^{d_2\times d_1},\ x\in\mathbb R^{d_1}, $$ LoRA adds a trainable low-rank adapter $$ y=(W+\Delta W)x+b,\qquad \Delta W=\gamma_r\,B\,A, $$ where $$ B\in\mathbb R^{d_2\times r},\qquad A\in\mathbb R^{r\times d_1}. $$ The original LoRA choice is $$ \gamma_r=\frac{\alpha}{r}. $$ Kalajdzievski studies the limit $r\to\infty$ and defines “rank-stabilized” to mean that both activations and gradients remain $\Theta_r(1)$. Under initialization $B_0=0$, $A_0\sim\mathcal N(0,\sigma_A^2)$ iid, and adapter $$ f(x)=\gamma_r\,B\,A\,x, $$ the only choice of $\gamma_r\to0$ as $r\to\infty$ that makes all forward-pass moments $\mathbb E[\|f(x)\|^m]$ and all backward-pass moments $\mathbb E[\|\nabla_x\mathcal L\|^m]$ remain $\Theta_r(1)$ for every fixed $m$ is $$ \gamma_r\propto \frac{1}{\sqrt r}. $$ The derivation uses the expansions $$ B_n=-\eta\,\gamma_r\sum_{k