Papers

We refute the Strong Roberson Conjecture, which asserts that adding any graph outside a class closed under minors and disjoint unions strictly increases the distinguishing power of homomorphism counts from that class. More precisely, we construct connected graphs $H$ for which counts from graphs excluding $H$ as a minor determine the number of homomorphisms from $H$ to any target graph. We also refute the analogous conjecture with immersions in place of minors. We give explicit infinite families of excluded graphs, including cubic bipartite graphs that yield counterexamples for both relations. The proof introduces a method for deriving exact homomorphism count dependence from modular equivalences. We obtain these equivalences for infinitely many primes using prime-order automorphisms of graphs that exclude their orbit quotients as minors or immersions.

We present several geometric constructions of graphs with rapidly growing chromatic numbers, most of which are triangle-free or have large girth.

Given a graph $G$ and a positive integer $k$, a packing $k$-domatic coloring of $G$ is a function $f: V(G) \to \{1,\ldots,t\}$ such that (1) for any color $j \in \{1,\ldots, t\}$ and any two distinct vertices $u,v\in V(G)$ with $f(u)=f(v)=j$ we have $d_G(u,v)>j$, and (2) $V(G)$ admits a partition into $k$ sets $A_1,\ldots,A_k$ such that for any $v\in V(G)$ and any $i\in\{1,\ldots,k\}$ there exists a vertex $w\in A_i$ such that $d_G(v,w)\le f(w)$. The minimum integer $t$ such that $G$ admits a packing $k$-domatic coloring of $G$ using colors in $\{1,\ldots,t\}$ is denoted by $\chi_{\rho,k}(G)$. The condition (1) in the definition of a packing $k$-domatic coloring implies that $G$ is a packing coloring, hence $\chi_{\rho,k}(G)\ge \chi_\rho(G)$ holds for any graph $G$, where $\chi_\rho(G)$ is the packing chromatic number of $G$. On the other hand, the new concept also leads to a generalization of the domatic number of a graph due to which one can easily see that $\chi_{\rho,2}(G)=\chi_\rho(G)$ holds for any graph with no isolated vertices. One of the main result in this paper is that $\chi_{\rho,3}(G)=\chi_\rho(G)$ holds in any connected graph $G$ with minimum degree at least $2$, and the bound is best possible in two different senses. In addition, we provide several exact values and bounds on the new invariant in paths and cycles. We prove that $\chi_{\rho,k}(P_n)=k$ for any $k\in\{3,4,5\}$ as soon as $n\ge 8$, and, in contrast, $\chi_{\rho,k}(P_n)>k$ for any $n\ge k\ge 12$. We also prove the exact values of $\chi_{\rho,k}(P_\infty)$ when $k\in \{3,4,5\}$ for the two-way infinite path $P_\infty$, and exact values of $\chi_{\rho,k}(C_n)$ when $k\in\{3,4\}$ for all cycles $C_n$. We also consider a general framework of $S$-packing $k$-domatic colorings, where $S$ is an arbitrary sequence of non-negative integers, and present some basic results in this context.

In the generalization of token graphs (or supertoken graphs), the vertices of a $k$-token graph of a graph $G$ correspond to $k$ distinguishable or indistinguishable tokens placed on $k$ different or equal vertices of $G$. In several cases, we provide the whole spectrum of a supertoken graph of a star by analyzing its regular partitions.

We give a complete classification of Schur positivity for three-legged spiders. Every spider with at least two even legs is Schur positive, whereas every spider with three odd legs has a negative Schur coefficient. If $a$ is even and $b,c$ are odd, then the spider $S(a,b,c)$ is Schur positive if and only if $a\le 5b+5c+2$. Whenever Schur positivity fails, there is exactly one negative Schur coefficient, whose value we determine explicitly. These results extend those of Thibon and Wang for the families $S(a,2,1)$ and $S(a,4,1)$, and those of Wang and Wang for $S(a,b,2)$. The proof combines known Schur-positivity results for clique-spiders with Pieri's rule and simultaneous induction.

We extend the partition pairing theory of Andrews and Dastidar by replacing pairs with groups of $k$ equal parts. Two weight-preserving bijections give combinatorial interpretations of the joint pairing index–width distribution and the negative-rank enumerations. The first combines conjugation with the Stockhofe–Keith correspondence and sends the $k$-pairing index and width to the number of parts and the largest part, respectively. Consequently, their joint distribution is independent of $k$ and is given by a Gaussian polynomial. We also obtain finite refinements that record the residual statistics. The second bijection sends simply $k$-paired partitions of negative $k$-pairing rank to $k$-regular partitions with marked internal gaps. In a fixed nonzero residue class, sign cancellation leaves rectangular partitions, while gap markings correspond to overlining choices with the smallest part not overlined. This gives direct combinatorial explanations of the divisor counts and the factor $1/2$ in the overpartition enumeration, extending the odd-divisor and odd-overpartition results of Andrews and Dastidar. Finally, motivated by the diagonal pairing of Andrews and Dastidar, we extend the construction to ordered tuples of $k$ Young diagrams. Applying the two-wing transfer operation to pairs of component diagrams defines an equivalence relation on these tuples. We show that two tuples are equivalent if and only if they have the same cell-multiplicity function, and that every equivalence class contains a unique representative whose component diagrams are nested. We also determine the cardinalities of the equivalence classes and identify the nested representatives with plane partitions of rectangular shape.

Bent partitions were introduced as a generalization of partial spread construction of bent functions and they became a hot research topic recently. In this paper, we prove two general constructions of normal bent partitions that are related to vectorial dual-bent functions. They cover many of the currently known bent partitions of this type as partial cases. These constructions also provide a large number of new bent partitions. Relevant properties of bent functions obtained from such partitions are proven.

The canonical Ramsey theorem establishes that for every positive integer $t$, there is a sufficiently large $n$ such that every edge-colouring of a complete graph $K_n$ contains a copy of $K_t$ that is canonically coloured, i.e. monochromatic, rainbow, or lexicographic. In this paper we investigate two variations of this theorem. First, we study canonical Ramsey numbers in the multipartite hypergraph setting. We prove that in every edge-colouring of the 3-uniform complete hypergraph on at least $t^{151t^2}$ vertices, there always exists a canonically coloured copy of $K_{t, t, t}^{(3)}$. This estimate is sharp up to the constant in the exponent. Second, we explore off-diagonal canonical Ramsey numbers. Let $ER(a, b, c)$ denote the minimum number of vertices $n$ required to guarantee a monochromatic $K_a$, a lexicographic $K_b$, or a rainbow $K_c$ in every edge-colouring of the complete graph $K_n$. We establish sharp bounds for these numbers across different parameter regimes. Specifically, we prove that $ER(a, b, c)\geq c^{\Omega(ab)}$ when $c$ is sufficiently large relative to $a$, and that $ER(a, b, c)\leq 2^{O_b(a)}$ when $b\ge 4$ is a fixed constant and $c\le a$. Finally, we analyse the behaviour of the function when avoiding lexicographic triangles (i.e. $b=3$), showing that $ER(a, 3, c)\le (a-1)(c-2)+O(c^6)$.

Inspired by work of Gaster, Greene, and Vlamis on the chromatic number of curve graphs, we prove that every endomorphism of the total Kneser graph is an automorphism. We also show that the automorphism group of the total Kneser graph is the product of a pair of permutation groups.

Let $\Palt=(\C,\T)$ be a $k$-palette. For $0\le t\le k-1$, its $t$th coordinate-extension degree is the minimum, over every choice of $t$ coordinates and every assignment of colors to them, of the proportion of assignments to the remaining $k-t$ coordinates that complete the fixed colors to an admissible $k$-tuple. For a $k$-graph $F$, we define $\pi_t^{\ext}(F)$ as the supremum of this degree over all palettes not admitted by $F$. We prove that \[ \pi_t^{\ext}(F)=0 \quad\text{if and only if}\quad F\text{ is }t\text{-layered}. \] We also relate $t$-layeredness to vanishing orders, min-layeredness, max-layeredness, and layeredness. These results recover and extend previous characterizations of Reiher, Rödl, and Schacht and of Lamaison, and answer a question of Lamaison for $3$-graphs. At $t=0$, the parameter $\pi_0^{\ext}(F)$ is the $(k-2)$-uniform Turán density $\pi_{k-2}(F)$. For every $k\ge3$ and $r\ge2$, we construct a finite $k$-graph $F_{k,r}$ with \( \pi_{k-2}(F_{k,r})=2(r-1)/rk^k. \) Thus $2/k^k$ is an accumulation point for single forbidden $k$-graphs. We also show that the least density of a $k$-graph that fails condition $\Sp$ of Lin, Wang and Zhou is $4/(3k^k)$. Finally, for every admissible matching of size $m$, we construct a $k$-graph that satisfies $\Sp$ for every coordinate pair, has no vanishing order, and has density $2^m/k^k$. This disproves a conjecture of Lin, Wang and Zhou for every $k\ge3$.

For every fixed $r\ge3$ and all sufficiently large $n$, we determine the largest adjacency spectral radius of an $n$-vertex graph with fewer than $q c_r(n)$ copies of $K_{r+1}$, where $1\le q<n/r$. Here $c_r(n)$ is the number of copies created by adding one edge to a largest part of the Turán graph. We also determine all extremal graphs. In most cases the extremal graph is obtained from an almost balanced complete multipartite graph by adding a star in one part. Two small values of $q$ require separate constructions, and an additional transition occurs when $r=3$ and $n\equiv2\pmod3$. Under the non-strict constraint, the unique extremal graph is obtained by adding a $q$-edge star to a largest part of the Turán graph. We determine the difference between the strict and non-strict values and prove that the sharp matching threshold is $\sqrt2(r-1)/r$. The proof treats separately the ranges $q=o(n)$, $q/m\to\gamma\in(0,1)$, and $q/m\to1$.

The famous and actively studied problem of Brown–Erdős–Sós from 1973 asks for $f^{(r)}(n;s,k)$, the maximum number of edges in an $r$-graph with $n$ vertices in which no $s$ vertices span $k$ or more edges. In this paper, we concentrate on the case $r=3$ and $s=k+2$, with $k\ge2$ fixed and $n\to\infty$; then it is easy to show that the extremal function grows quadratically in $n$. Delcourt and Postle proved that the limit $\pi(k):=\lim_{n\to\infty} f^{(3)}(n;k+2,k)/n^2$ exists for every $k$. While Brown, Erdős and Sós observed that $\pi(2)=1/6$ already in the 1970s, the value of $\pi(k)$ for $3\le k\le 7$ was determined only recently (by various subgroups of Glock, Joos, Kim, Kühn, Lichev, Pikhurko, and Sun). Very recently, Chao, Huang and Liu determined $\pi(k)$ for every odd $k$. Independently of the last result, we show that $\pi(9)=1/5$. Also, we prove that $\pi(8)\le {5053}/{26544}$, which is within $0.0029$ of the best known lower bound $\pi(8)\ge 3/16$. The new upper bounds are obtained by expressing some previous arguments as a linear program and then using a computer to generate and solve its instances. Our proof of the lower bound on $\pi(9)$ is based on a finite field construction combined with existing packing results.

We compute the singular cohomology groups of odd biisotropic symplectic Grassmannians $\operatorname{S_2Gr}(k, 2n+1)$ by presenting an explicit dimension-preserving combinatorial bijection between the Bialynicki-Birula cells in $\operatorname{S_2Gr}(k, 2n+1)$ and the Schubert cells in $\operatorname{Gr}(k, 2n+2-k)$.

A $4$-general set of PG(n,q) is a point set with no four coplanar,and Mb{n}{q} is the largest such size. The climb problem asks, at each $n$, whether the constructions of Pavese (2025) can be improved by one point. We show that the answer is governed by the code tables: $PG(M-1,q)$ contains an $N$-point $4$-general set exactly when a projective $[N,N-M,\ge 5]_q$ code exists. Over $F_4$ this settles the first two rungs: Mb{3}{4}=5 and Mb{4}{4}=11, each with two independent proofs, so Pavese's constructions are optimal at both. Over $F_5$ the same reduction gives Mb{4}{5}=12, one more than the near-MDS lower bound. At $n=5$ the tables fall silent; we state the verified position $21\le Mb{5}{4}\le 30$ rather than the tabulated $\le 29$, which rests on a private communication. Any $22$-point $4$-general set of $PG(5,4)$, if one exists, shares at most $12$ points with Pavese's extremal $21$-point set; the proof of this rigidity statement is a finite computation, with certificates in Appendix~A.

In 2012, Mirtabatabaei and Bullo studied heterogeneous Hegselmann–Krause models of opinion dynamics, where different agents may have different confidence or influence bounds. They conjectured that opinion vectors always converge, a fundamental problem that remains open. In support of this main conjecture they gave some partial results and made some auxiliary conjectures. Here we give counterexamples to two of these auxiliary conjectures, and even to one theorem in their paper. We explain why these disproved conjectures and false theorem nevertheless likely remain valid for almost all initial opinion vectors. In the last section (Section 7), we describe the role of AI in the production of the results in this paper.

We study two variants of location-domination in infinite triangular grids which allow locating up to $\ell$ vertices simultaneously. A locating-dominating code $S$ is a dominating vertex set such that every vertex outside of it has a unique neighborhood within the code $S$. As this neighborhood is unique, we may use it as an identifier for locating the considered vertex outside of the vertex set. With the variants we study, we may also locate sets of vertices instead of only single vertices. In particular, we obtain the exact minimum densities of these types of sets in the infinite triangular grid for every value of $\ell$.

We determine the orientable genus of an infinite family of bipartite Kneser graphs. The graph $H(h,2)$ has two copies of the two-element subsets of $[h]$, with opposite-class vertices adjacent when the corresponding subsets are disjoint. For every prime $h>3$ with $h\equiv3\pmod8$, we prove $$ \gamma(H(h,2))=1-\frac{h(h-1)}{2} +\frac{h(h-1)(h-2)(h-3)}{16}. $$ Euler's formula gives this lower bound, with equality for a quadrangulation. We construct a vertex-transitive orientable quadrangulation using an odd-order affine group that acts simply transitively on the two-element subsets. This gives an infinite family satisfying Pisanski's conjecture on quadrilateral embeddings of regular bipartite graphs.

We study the actions of hyperoctahedral groups on the cohomology of the minimal De Concini-Procesi wonderful models associated with reflection arrangements of type $B$. We express the generating series of their equivariant Poincaré polynomials in terms of the corresponding type $A$ series and the equivariant reduced characteristic series of type $B$. The key ingredient is a general inversion formula in equivariant incidence algebras relating equivariant Chow functions and super-reduced characteristic functions. Applying this formula to partition and signed partition lattices, we recover Getzler's compositional identity in type $A$ and establish its type $B$ analogue. We also obtain explicit plethystic and infinite-product formulas for the equivariant characteristic series of type $B$.

A set $S$ of vertices in a graph $G$ is a dominating set of $G$ if every vertex not in $S$ has a neighbor in $S$, where two vertices are neighbors if they are adjacent. The domination number, $\gamma(G)$, of $G$ is the minimum cardinality among all dominating sets of $G$. Given a set $S$ of vertices of a graph $G$, two vertices are located by $S$ if they have distinct sets of neighbors in $S$. Moreover, if $S$ locates every pair of vertices not in $S$, then it is called a locating set of $G$. A locating dominating set of $G$ is both a dominating and a locating set of $G$. The locating domination number, $\gamma^{\rm LD}(G)$, is the minimum cardinality among all locating dominating sets of $G$. A notable conjecture in the study of locating dominating sets is to show that the locating domination number of an isolate-free and twin-free graph of order $n$ is at most $\frac{1}{2}n$. So far, the best approximation to this upper bound conjecture is known to be $\left \lceil \frac{5}{8}n \right \rceil$. Much in line with the conjecture, an even stronger reformulation proposed in the literature asks if it is possible to partition the vertex set of an isolate-free and twin-free graph into two locating sets. However, such partitions into locating sets may not exist if the graph is also allowed to have twins. Continuing with this line of research, we show that if $G$ is an isolate-free (and not necessarily twin-free) graph, then the vertex set of $G$ can be partitioned into a dominating set and a locating dominating set. As a consequence, we infer that every isolate-free graph $G$ of order $n$ satisfies $\gamma(G) + \gamma^{\rm LD}(G) \le n$, and we show that the last bound is tight. Moreover, our proof of the existence of a partition of the vertex set of an isolate-free graph into a dominating and a locating dominating set also provides a polynomial-time algorithm to construct such a partition.

We prove Ehrhart positivity for series-parallel matroids and extend this result to all series-parallel extensions of multi-path matroids. A formula for the Ehrhart coefficients of series-parallel matroids is derived and used to prove strict positivity. An extension criterion based on weighted lattice-point counts is established. Recursive formulas for lattice-point counts are derived, and the recursion is shown to preserve a class of polynomials with nonnegative coefficients. We finally construct a connected transversal matroid with a negative Ehrhart coefficient, disproving the conjecture of Ferroni, Morales, and Panova that every connected matroid whose base polytope admits a subdivision into series-parallel matroid base polytopes is Ehrhart positive.

In this paper, we first expand fixed diagonals of the Jacobi-Stirling numbers in a binomial basis. For the second kind, the expansion coefficients are polynomials in $z+1$ with nonnegative integer coefficients. We give a recurrence and a signed-partition interpretation for these coefficients. The same holds for the differences between corresponding unsigned first-kind and second-kind coefficients. We then prove that every nonzero nonnegative linear combination of the descent polynomials over Jacobi-Stirling permutations with a fixed number of deleted barred letters has only simple negative zeros when that number is one, two, or three. The same holds when exactly one or two barred letters are retained. Thus we verify five infinite families in a conjecture of Gessel, Lin and Zeng. Finally, using insertion operators, we find that every descent polynomial over Jacobi-Stirling permutations with a fixed number of deleted barred letters is top heavy and has an increasing left half.

Every monotone Boolean function on the continuous cube admits a coalition of $O(n/(\varepsilon\log n))$ coordinates that can force a fixed output (either zero or one) with probability at least $1-\varepsilon$. This old conjecture of mine was recently proven by Chattopadhyay and Gurumukhani~\cite{CG}. This writeup contains a simplification of their proof, generated by AI after suggesting the use of the junta theorem of~\cite{F98} and the discretization procedure of ~\cite{F04}.

Let $a(n)$ be the number of standard Young tableaux of shape $(n,n,n)$ whose entry in row $2$, column $1$ is odd; equivalently, the number of those whose first descent is even. This is entry A011553 of the On-Line Encyclopedia of Integer Sequences, contributed in 1996, and after thirty years of curation it carries no formula. We supply one, $a(n) = 8\,\bigl(n!\,(n+2)!\bigr)^{-1}\sum_{m=1}^{\lfloor n/2\rfloor} m(m+1)(3n-2m-1)!/(n-2m)!$, and use it to settle both of the conjectures the entry records. A creative telescoping certificate shows that $a$ satisfies a linear recurrence of order two with polynomial coefficients; the order-three recurrence conjectured by R. J. Mathar in 2023 is a left multiple of it, with explicit cofactor $(4S^{-1}-3)/(7n-9)$. We also prove $a(n)\sim 3^{3n+7/2}/(64\pi n^{4})$, the asymptotic conjectured by V. Kotesovec in 2014. The second proof gives slightly more than the conjecture asks: the position of the first descent has a limiting distribution, the probability that the $(2,1)$ entry equals $r+1$ tending to $r(r+2)/3^{\,r+1}$. Summing the even terms, a uniformly random tableau of shape $(n,n,n)$ has an odd $(2,1)$ entry with probability tending to $27/64$, whereas for a uniformly random tableau of $n$ cells of unrestricted shape the corresponding limit is $1/e$.

The well-known inequality of Fiedler ensures that the spectral gap $r-\lambda_2(G)$ of a connected non-complete $r$-regular graph $G$ is bounded from above by the vertex connectivity $\kappa(G)$ of $G$. We prove that, for integers $t\geq2$ and $r>2t^2$, every connected $r$-regular graph with vertex connectivity at most $2t$ has spectral gap at most $\frac{1}{2}(r+t+2-\sqrt{(r-t+2)^2-4t(t-1)})$. We show that this bound is nearly optimal for each fixed $t \ge 2$ and sufficiently large $r > 2t^2$ such that $r+1$ is divisible by $t$. We further prove that if $r>(2t-1)(2t-2)$, then every connected $r$-regular graph whose vertex connectivity is odd and no more than $2t-1$ has spectral gap at most $r-\min\bigl\{\xi(r,t),\nu(r,t)\bigr\}$ for some explicitly given functions $\xi(r,t)$ and $\nu(r,t)$. If $r\geq6t^2$, then this minimum equals $\xi(r,t)=\frac{r(2r-4t+5)}{2(r-t+2)}$. In particular, when $\kappa(G)=2t-1$, this gives $r-\lambda_2(G)<t=(\kappa(G)+1)/2$, which improves Fiedler's bound by nearly a factor of two. Our upper bounds in their parameter ranges improve two known bounds on the spectral gap of regular graphs.

Random sampling in DNA sequencing produces repeated reads, increasing retrieval latency and sequencing cost. We study the coverage-depth problem for full-message recovery in distributed DNA storage under noiseless uniform sampling, where strands are partitioned among $M$ containers and one strand is independently sampled with replacement from each container per round. For arbitrary linear codes and ordered partitions, we derive exact formulas for the recovery-time distribution and expectation. We prove that MDS codes, whenever they exist, are optimal for every fixed partition, and establish a universal lower bound on the expected total read cost together with its equality conditions. For MDS codes, we identify container-size regimes that yield genuine savings in total reads and regimes that provide only parallelism without changing the asymptotic sequencing cost. For simplex codes, we prove that the $q$-ary simplex code is, up to isomorphism, the unique single-container minimizer among codes with the same parameters, resolving a recent conjecture by Bertuzzo, Ravagnani, and Yaakobi. We further construct a partition attaining the minimum total read cost and derive bounds for intermediate and balanced partitions. These results clarify when distributed sampling reduces latency alone and when it also reduces sequencing cost.

We establish local and global spectral bounds for Hermitian $A_\alpha$-matrices of mixed graphs. Using the first three spectral moments, we obtain an upper bound for the largest eigenvalue as the largest real zero of an explicit cubic polynomial. Vertexwise estimates for the extreme eigenvalues yield new lower bounds for the spectral spread, including a bound that strictly improves an existing degree-based estimate for ordinary graphs. A local two-dimensional compression produces a spread bound involving neighbour-degree data and gain-weighted triangles. This bound is exact for every mixed orientation of a star and is independent of a known Zagreb-index bound. We also derive two complementary upper bounds for sums of the smallest eigenvalues. As further consequences, the spectral estimates provide a computable convergence guarantee for Richardson graph filtering and a stability certificate for residual graph-neural-network layers on directed networks. Numerical examples illustrate the sharpness and mutual incomparability of the proposed bounds.

For a tree $T$, let $g(T)=\lambda_1(T)-\lambda_2(T)$ be the difference between its two largest adjacency eigenvalues. A balanced double comet is obtained by attaching equally many leaves to the two endpoints of a path. Jovović, Koledin and Stanić conjectured that such a tree attains the minimum adjacency spectral gap among trees of any fixed order. We prove that every minimizing tree of order $n\ge200$ is a balanced double comet. We also show that, for any finite irreducible reversible continuous-time Markov chain, the inverse spectral gap differs from the effective resistance between two states times the stationary variance of their hitting probability by at most the inverse Dirichlet gap for killing at those states. For Perron chains, we give an exact two-vertex Schur-complement formula for this resistance–variance quantity.

We prove both Erdős-Ulam conjectures on monochromatic union-closed families. Every two-colouring of the subsets of a finite set contains a monochromatic union-closed family whose size grows faster than any fixed power of the size of the set, while suitable colourings admit no monochromatic union-closed family of exponential size. Both results hold for any number of colours and are verified in Lean.

Let $n$ be even, let $\zeta$ be a primitive $n$-th root of unity, and let $\Delta_n$ be the set of derangements in the affine group $\overrightarrow{GL}(\mathbb{Z}/n\mathbb{Z})$. We evaluate the Wang–Sun type sum \[ S_n=\sum_{\pi\in\Delta_n}\mathrm{sign}(\pi) \prod_{x\in\mathbb{Z}/n\mathbb{Z}} \frac{1+\zeta^{x-\pi(x)}}{1-\zeta^{x-\pi(x)}}. \] The main step is an arithmetic grouping: for an affine map $\pi(x)=vx+u$, the product depends only on $\sigma=\gcd(1-v,n)$ and on the residue class of $u$ modulo $\sigma$. After summing the translation signs, the full sum becomes a sum over even divisors with explicit local multiplicities. The only remaining analytic blocks are alternating even cotangent power sums; a recent formula of Liu and Xin then gives a closed expression in Bernoulli polynomials and universal coefficients. Moreover, using results by Cvijović and Klinowski we can evaluate the cotangent power sums using exclusively rational arithmetic. This supplies a general formula for the full affine sum whose initial values were computed using the definition only in an earlier work. The formula also lets us prove some general patterns of the sum: it has integral value for powers of $2$, prime denominator for $2p$ for primes $p>3$ and it has alternating signs.

We prove the following conjecture of Zhang (arXiv:2604.10779, Conjecture 6.1): for $n \geq 0$, the number of $2$-stack-sortable permutations of $\{0,1,\dots,n\}$ ending in $0$ is the $n$th Motzkin number. By Zhang's result, there is a bijection between $2$-stack-sortable permutations ending in their least element and standard composition tableaux of width at most $2$. We then show bijectively that there are an equal number of these and standard Young tableaux of width at most $3$, which are known to be counted by the Motzkin numbers.