Introduction

Ribbon functions connect Boolean inclusion–exclusion, descent representations, and rank-selected homology. For a nonempty composition $\alpha \models n$, the ribbon element in the algebra of noncommutative symmetric functions is $$R_{\alpha }=\sum _{\beta ⪰\alpha }{(-1)}^{\ell (\alpha )-\ell (\beta )}{H}_{\beta }.$$ Under abelianization, $R_{\alpha }$ maps to the ribbon Schur function $r_{\alpha }$, which is the Frobenius characteristic of the corresponding descent representation. We refine this picture by two cycle statistics and then study the zigzag compositions that encode alternating fences.

For $m\ge 0$, write $$m^{+}=\lceil \frac{m}{2}\rceil ,m^{-}=\lfloor \frac{m}{2}\rfloor ,$$ and set $${𝗁}_m(t,q)=\frac{1}{m!}\prod _{i=1}^{m^{-}}(t+i)\prod _{j=0}^{m^{+}-1}(t+jq).$$ The second product is the cycle enumerator of a permutation in ${𝔖}_{m^{+}}$, with $q$ recording its reflection length $m^{+}-c(\sigma )$. The first product is obtained from the cycle enumerator of a permutation on $m^{-}+1$ letters with one distinguished letter. Since the complete functions freely generate $\mathrm{NSym}$, the assignments $H_m\mapsto {𝗁}_m(t,q)$ define an algebra homomorphism ${\varphi }_{t,q}$. We write $$Z_{\alpha }(t,q)={\varphi }_{t,q}(R_{\alpha }).$$

Main results

Our first result gives an equivariant homological realization of the ribbon evaluation. The chain groups are spanned by ordered set partitions whose blocks carry an ordinary permutation and a rooted permutation. Adjacent blocks are merged by direct sum in the ordinary component and by gluing the two root cycles in the rooted component. Suppose that $\alpha$ has at most one odd part. For a total decoration $\theta$, let $F_{\alpha }(\theta )$ be the set of cuts at which both components factor. The fiber with total decoration $\theta$ is the classical ribbon complex with cut set $F_{\alpha }(\theta )$. Consequently, theorem 3.11 gives $$\begin{array}{ccc}& {\mathrm{ch}}_{t,q}{H}_k(C_{\bullet }^{t,q}(\alpha ))=\sum _{\begin{array}{c}\theta \in {𝒟}_n\\ |F_{\alpha }(\theta )|=k-1\end{array}}{t}^{{\mathrm{deg}}_t\theta }{q}^{{\mathrm{deg}}_q\theta }{r}_{{\gamma }_{\alpha }(\theta )}.& \text{(1)}\end{array}$$ Thus every bigraded homology representation is determined explicitly and is Schur-positive. Grouping by exact cut set gives the nonnegative multiplicity polynomials in corollary 3.12.

The same factorization data define a cut coproduct. After the complexes are organized as a species on finite label sets, the coproduct makes their direct sum a counital differential graded comonoid in the Cauchy monoidal category. It respects the total-decoration fibers. At every factorization cut, the aggregate label-set component induces a canonical injection of the corresponding ribbon homology module into the induction product of the two factor modules; the cokernel has the near-concatenation ribbon character. The kernel on homology of the aggregate reduced coproduct is exactly $H_1$; see theorem 3.14, corollary 3.15, and corollary 3.16.

Our second result concerns the zigzag compositions ${\delta }_{2N}=(2,\dots ,2)$ and ${\delta }_{2N+1}=(2,\dots ,\mathrm{2,1})$. At $q=-1$, their evaluations are the order polynomials of alternating fences. Retaining reflection length by writing $q=-u$, theorem 4.3 proves $$\begin{array}{ccc}& n!Z_{{\delta }_n}(t,-u)\in \mathbb{N}[t,u].& \text{(2)}\end{array}$$ The proof starts from the hypergeometric differential equation for the even kernel and converts it into two simultaneous recurrences with nonnegative coefficients. The odd case follows from a first-order relation between the even and odd kernels.

The same total decoration controls the two parts of the paper. Its factorization cuts select the ribbon complex occurring in homology, while the excess of ordinary cycles over the factorization intervals determines the sign after the substitution $q=-u$. More precisely, if $\theta =(\sigma ,\tau )$ contributes in homological degree $k$ and $q$-degree $j$, then $${\mathrm{fd}}_n(\theta )=c(\sigma )-|F_{{\delta }_n}(\theta )|-1=\lceil \frac{n}{2}\rceil -k-j\ge 0.$$ The differential preserves total decoration and therefore splits the zigzag complex by this factorization defect. On each defect layer the shifted Euler sign is constant, and theorem 4.5 gives the support condition $$H_k^{i,j}(n)\ne 0⟹k+j\le \lceil \frac{n}{2}\rceil \le i+j.$$ The defect-zero edge of this region is computed in corollary 4.7 by explicit representation-valued formulas. The virtual Euler characteristic is not Schur-positive in general, already for $n=4$, so the scalar positivity in (2) does not arise from coefficientwise positivity in the representation ring.

The Riccati recurrence also gives positive integral logarithmic coefficients for the even zigzag series. At $u=1$, they specialize to scaled order polynomials of alternating circular fences; see proposition 4.8.

Relation to earlier work

Up to augmentation and homological reindexing, the undecorated complex is the ordered-set-partition complex ${\mathrm{\Delta }}_{\alpha }$ of Ehrenborg and Jung. They identify it with a rank-selected Boolean complex, prove concentration of its homology, construct a basis indexed by exact descent composition, and identify the top homology with the border-strip Specht module (Ehrenborg and Jung 2013, Theorem 4.2, Theorem 6.6, and Proposition 7.3). Bergeron and Krob constructed acyclic complexes related to ribbon elements in noncommutative symmetric functions (Bergeron and Krob 1997). After choosing an $n$-dimensional vector space with an ordered basis, the multidegree $(1,\dots ,1)$ part of VandeBogert’s refinement complex for its symmetric algebra has the same ordered-set-partition basis and adjacent multiplication maps, up to homological reindexing (VandeBogert 2025, Definition A.7 and Example A.8). In the present complex, these classical ribbon complexes occur as fibers indexed by total permutation decorations, and the simultaneous factorization cuts determine both the fiber and its bidegree.

Chain-level categorifications of the ribbon concatenation and near-concatenation identity occur in VandeBogert’s ribbon Schur functors and, for projective $0$-Hecke modules, in the canonical split exact sequences of Almousa and Lu (VandeBogert 2025; Almousa and Lu 2026). Here the injection in corollary 3.15 is induced by a coassociative label-set cut coproduct on ${𝔖}_n$-equivariant complexes. It restricts to each total-decoration fiber and is compatible with simultaneous factorization at every cut. Hersh and Sundaram construct ribbon bases for rank-selected homology and Whitney homology of geometric lattices (Hersh and Sundaram 2026, Theorems 5.29–5.30).

Gessel and Reutenauer jointly enumerate cycle structure and descent sets of a single permutation (Gessel and Reutenauer 1993). Novelli, Thibon, and Toumazet lift the cycle-index polynomials of symmetric groups to bases of quasisymmetric and noncommutative symmetric functions and obtain a product formula and a combinatorial-complex recurrence (Novelli et al. 2020). In our construction, cycle statistics are carried by the block decorations, while the descent set is supplied by the homology of the ribbon fiber. Direct-sum factorization of ordinary permutations is standard; see (Hivert et al. 2008, Equation (34), Proposition 3.3, and the discussion before Theorem 4.7). The construction below additionally uses a rooted component and requires simultaneous factorization of the two components at the same cuts.

Kreweras’s determinant for skew plane partitions (Kreweras 1965), in the form used by Ferroni, Morales, and Panova, gives the alternating-fence specialization below. Ferroni, Morales, and Panova prove coefficientwise positivity for fence and circular-fence order polynomials (Ferroni et al. 2025), and Kahane gives a permutation statistic for the coefficients of every fence order polynomial (Kahane 2026, Theorem 3.8). The author’s earlier preprint gives a different record model for the same one-variable specialization and for circular fences (Huang 2026). These results concern $u=1$; the refinement here retains reflection length and relates its sign to the homological factorization defect.

Organization and conventions

Section 2 defines the two-variable ribbon evaluation. Section 3 constructs the decorated complex and computes its equivariant homology. Section 4 proves the alternating-fence specialization, bivariate positivity, the defect decomposition, the support and defect-zero formulas, and the logarithmic consequence.

All vector spaces are over a field $𝕂$ of characteristic zero, and $\mathbb{N}$ includes zero. For $m\ge 0$, write $[m]=\{1,\dots ,m\}$, with $[0]:=⌀$; an integer interval $\{a,\dots ,b\}$ is empty when $a>b$. Polynomial characters take values in $\mathbb{Q}[t,q]$. Chain complexes are homologically graded. The superscript $t,q$ records the two weight gradings; the differential does not depend on numerical values of $t$ and $q$. Compositions in section 3 are nonempty. Empty internal alphabets and cut sets are allowed. Throughout, ${𝔖}_n$ acts on the labels of ordered set partitions and fixes the decoration bases.

Ribbon evaluations weighted by cycle statistics

The ribbon basis turns coarsenings into inclusion–exclusion. We evaluate the complete generators by two cycle enumerators.

Compositions, ribbons, and $\mathrm{NSym}$

A composition of a positive integer $n$, denoted $\alpha \models n$, is a sequence $\alpha =({\alpha }_1,\dots ,{\alpha }_k)$ of positive integers with $$|\alpha |:={\alpha }_1+\cdots +{\alpha }_k=n.$$ Its length is $\ell (\alpha ):=k$. We also use the empty composition $⌀\models 0$, of length zero.

A composition $\gamma$ is a coarsening of $\alpha$ if it can be obtained from $\alpha$ by repeatedly replacing adjacent parts by their sum. Write $\mathrm{Coars}(\alpha )$ for all coarsenings of $\alpha$, including $\alpha$.

Let $ℛ:=\mathbb{Q}[t,q]$. The algebra of noncommutative symmetric functions over $ℛ$ is the free associative algebra $${\mathrm{NSym}}_{ℛ}=ℛ⟨H_1,H_2,\dots ⟩,$$ where $H_0:=1$ and $\mathrm{deg}{H}_m=m$. For $\alpha =({\alpha }_1,\dots ,{\alpha }_k)$, set $H_{\alpha }=H_{{\alpha }_1}\cdots H_{{\alpha }_k}$, with $H_{⌀}=1$. We use the ribbon normalization $$\begin{array}{ccc}& R_{\alpha }=\sum _{\gamma \in \mathrm{Coars}(\alpha )}{(-1)}^{\ell (\alpha )-\ell (\gamma )}{H}_{\gamma },R_{⌀}=1.& \text{(3)}\end{array}$$ These are standard conventions; see (Gelfand et al. 1995). Under the canonical abelianization $\mathrm{NSym}\to \mathrm{Sym}$, $R_{\alpha }$ maps to the ribbon Schur function $r_{\alpha }$.

The two-variable character

For $m\ge 0$, put $$m^{+}=\lceil \frac{m}{2}\rceil ,m^{-}=\lfloor \frac{m}{2}\rfloor .$$ Thus $m^{+}+m^{-}=m$. Define $$\begin{array}{ccc}& {𝗁}_m(t,q)=\frac{1}{m!}\left(\prod _{i=1}^{m^{-}}(t+i)\right)\left(\prod _{j=0}^{m^{+}-1}(t+jq)\right),{𝗁}_0(t,q)=1.& \text{(4)}\end{array}$$

The map exists uniquely because ${\mathrm{NSym}}_{ℛ}$ is freely generated by the $H_m$. Here a character is a unital algebra homomorphism, as usual in combinatorial Hopf algebra theory (Aguiar et al. 2006).

The unsigned Stirling cycle enumerator (Stanley 2012, sec. 1.3), after homogenization, gives for $r\ge 1$ $$\begin{array}{ccc}& \sum _{\sigma \in {𝔖}_r}{q}^{r-c(\sigma )}{t}^{c(\sigma )}=\prod _{j=0}^{r-1}(t+jq),& \text{(6)}\end{array}$$ where $c(\sigma )$ counts all cycles, including one-cycles. Consequently, $$\begin{array}{c}m!{\varphi }_{t,q}(H_m)=\left(\prod _{i=1}^{m^{-}}(t+i)\right)\sum _{\sigma \in {𝔖}_{m^{+}}}{q}^{m^{+}-c(\sigma )}{t}^{c(\sigma )}.\end{array}$$

Thus both factors in the definition have direct cycle interpretations, and $r-c(\sigma )$ is the reflection length of $\sigma$ with respect to all transpositions.

At $q=-1$, the two products in (4) combine to give the falling factorial ${(t+m^{-})}^{\underline{m}}$, where $x^{\underline{m}}=x(x-1)\cdots (x-m+1)$. Hence $$\begin{array}{c}{\varphi }_{t,-1}(H_m)=\left(\genfrac{}{}{0px}{}{t+m^{-}}{m}\right)=\left(\genfrac{}{}{0px}{}{t+\lfloor m/2\rfloor }{m}\right).\end{array}$$ Thus, writing ${\varphi }_t={\varphi }_{t,-1}$, $$\begin{array}{cccc}& {\varphi }_t(H_{2a})& =\left(\genfrac{}{}{0px}{}{t+a}{2a}\right),& \text{(7)}\\ & {\varphi }_t(H_{2a+1})& =\left(\genfrac{}{}{0px}{}{t+a}{2a+1}\right).& \text{(8)}\end{array}$$

Applying ${\varphi }_{t,q}$ to (3) gives $$\begin{array}{ccc}& Z_{\alpha }(t,q)=\sum _{\gamma \in \mathrm{Coars}(\alpha )}{(-1)}^{\ell (\alpha )-\ell (\gamma )}\prod _{j=1}^{\ell (\gamma )}{𝗁}_{{\gamma }_j}(t,q).& \text{(9)}\end{array}$$ Reversal bijects the coarsenings of $\alpha$ with those of ${\alpha }^{\mathrm{rev}}$, preserves their lengths, and reverses their parts. Since the scalar factors in (9) commute, $$\begin{array}{ccc}& Z_{{\alpha }^{\mathrm{rev}}}(t,q)=Z_{\alpha }(t,q).& \text{(10)}\end{array}$$

For later use, define the zigzag compositions $$\begin{array}{c}{\delta }_0=⌀,{\delta }_{2N}=(\underset{N\text{ parts}}{\underbrace{2,\dots ,2}}),{\delta }_{2N+1}=(\underset{N\text{ parts}}{\underbrace{2,\dots ,2}},1).\end{array}$$ By (10), the last composition may equivalently be written $(\mathrm{1,2},\dots ,2)$.

The decorated ribbon complex

Ordered set partitions realize the alternating sum over coarsenings at chain level. After recalling the undecorated ribbon complex, we establish the factorization criterion and apply it to rooted permutation decorations.

Throughout this section, $\alpha \models n$ is nonempty, so $n\ge 1$. All representations are finite-dimensional.

The classical ribbon complex

Thus the interval of coarsenings of $\alpha$ is canonically a Boolean lattice on the $r-1$ cuts in $D(\alpha )$. If $\beta =({\beta }_1,\dots ,{\beta }_k)⪰\alpha$ and $1\le i<k$, let $$m_i\beta :=({\beta }_1,\dots ,{\beta }_{i-1},{\beta }_i+{\beta }_{i+1},{\beta }_{i+2},\dots ,{\beta }_k).$$ Deleting one more cut shows that $m_i\beta ⪰\alpha$.

For $S\subseteq D(\alpha )$, let ${\gamma }_S$ denote the unique composition of $n$ with $D({\gamma }_S)=S$.

Ordered set partitions.

The stabilizer of the standard ordered partition is the corresponding Young subgroup. Hence, for every $\beta \models n$, $$\begin{array}{cccc}& |{\mathrm{OSP}}_{\beta }[n]|& =\left(\genfrac{}{}{0px}{}{n}{{\beta }_1,\dots ,{\beta }_k}\right)=\frac{n!}{{\beta }_1!\cdots {\beta }_k!},& \\ & M^{\beta }& \cong {\mathrm{Ind}}_{{𝔖}_{{\beta }_1}\times \cdots \times {𝔖}_{{\beta }_k}}^{{𝔖}_n}\mathrm{𝟏},\mathrm{ch}(M^{\beta })=h_{\beta }:=h_{{\beta }_1}\cdots h_{{\beta }_k}.& \text{(11)}\end{array}$$

The chain complex.

For $𝐁=(B_1,\dots ,B_k)\in {\mathrm{OSP}}_{\beta }[n]$, define $$d_i𝐁:=(B_1,\dots ,B_{i-1},B_i\bigsqcup B_{i+1},B_{i+2},\dots ,B_k)\in {\mathrm{OSP}}_{m_i\beta }[n].$$ Linear extension gives an ${𝔖}_n$-equivariant map $$d_i^{\beta }:M^{\beta }⟶M^{m_i\beta }.$$

Associativity of disjoint union gives the usual face identities $$d_i{d}_j=d_{j-1}{d}_i(1\le i<j\le k-1).$$ The two orders of deleting any pair of cuts therefore give equal maps with opposite signs in (12). Hence ${\partial }^2=0$.

Homology.

Let ${ℬ}_n$ be the Boolean lattice of subsets of $[n]$. For $D\subseteq [n-1]$, write $${ℬ}_n(D):=\{S\subseteq [n]:|S|\in D\}$$ for its proper rank-selected subposet.

Factorization along cuts

We now give a criterion that decomposes a decorated ribbon complex into subcomplexes with fixed total decoration. Fix a nonempty composition $\alpha \models n$. For each integer $m$ that occurs as a block size in a coarsening of $\alpha$, choose a finite set ${𝒜}_m$. Equip these sets with a bidegree $$\mathrm{bideg}:{𝒜}_m⟶{\mathbb{N}}^2,$$ whose coordinates we denote by ${\mathrm{deg}}_t$ and ${\mathrm{deg}}_q$. Suppose that whenever $a,b$ are adjacent block sizes in such a coarsening there is a product $$\diamond :{𝒜}_a\times {𝒜}_b⟶{𝒜}_{a+b}.$$ We require $$\mathrm{bideg}(x\diamond y)=\mathrm{bideg}(x)+\mathrm{bideg}(y)$$ and $(x\diamond y)\diamond z=x\diamond (y\diamond z)$ whenever the three consecutive block sizes occur in a coarsening of $\alpha$. Thus for every $\beta =({\beta }_1,\dots ,{\beta }_k)⪰\alpha$ there is an unambiguous total product $$\begin{array}{c}{\mu }_{\beta }:{𝒜}_{{\beta }_1}\times \cdots \times {𝒜}_{{\beta }_k}⟶{𝒜}_n,\end{array}$$ where ${\mu }_{(n)}={\mathrm{id}}_{{𝒜}_n}$.

Decorate an ordered set partition of type $\beta$ by one element of ${𝒜}_{{\beta }_i}$ on each block. Merging adjacent blocks and multiplying their decorations defines a complex $C_{\bullet }^{𝒜}(\alpha )$ with $$\begin{array}{c}{C}_k^{𝒜}(\alpha )=\underset{\begin{array}{c}\beta ⪰\alpha \\ \ell (\beta )=k\end{array}}{⨁}𝕂[{\mathrm{OSP}}_{\beta }[n]]\otimes 𝕂[{𝒜}_{{\beta }_1}]\otimes \cdots \otimes 𝕂[{𝒜}_{{\beta }_k}],\end{array}$$ with the signs of (12). The group ${𝔖}_n$ acts on the labels of the ordered set partition and fixes all decorations. Associativity gives the face identities and hence makes the boundary square to zero.

Condition (14) states that the factors at all selected cuts exist simultaneously and are unique.

For a finite-dimensional bigraded ${𝔖}_n$-module $V$, write $${\mathrm{ch}}_{t,q}V:=\sum _{d,e}\mathrm{ch}(V_{d,e})t^d{q}^e.$$ For a finite-dimensional bigraded vector space $V$, write $${\mathrm{Hilb}}_{t,q}V:=\sum _{d,e}\dim (V_{d,e})t^d{q}^e.$$

Permutation decorations

For $\sigma \in {𝔖}_a$ and $\tau \in {𝔖}_b$, define their direct sum $\sigma \oplus ︎\tau \in {𝔖}_{a+b}$ by $$(\sigma \oplus ︎\tau )(j)=\{\begin{array}{cc}\sigma (j),& 1\le j\le a,\\ a+\tau (j-a),& a<j\le a+b.\end{array}$$ Direct inspection gives the equalities $$\begin{array}{ccc}& (\sigma \oplus ︎\tau )\oplus ︎\upsilon =\sigma \oplus ︎(\tau \oplus ︎\upsilon ),c(\sigma \oplus ︎\tau )=c(\sigma )+c(\tau ).& \text{(17)}\end{array}$$

The decoration product.

The two components of ${𝒟}_m$ account for the two factors in (4): ordinary permutations record the $q$-weighted cycle statistic, while rooted permutations record the second cycle count.

For $s\ge 0$, let $${\Re }_s:={𝔖}_{\{\ast ,1,\dots ,s\}}$$ be the set of permutations of $s+1$ letters with distinguished letter $\ast$. We call its elements rooted permutations. The cycle containing $\ast$ has a unique cyclic notation $(\ast ,a_1,\dots ,a_u)$ beginning at the root. In particular, ${\Re }_0$ consists of the single rooted permutation $(\ast )$.

The sequences following the roots concatenate, and the nonroot cycles form a disjoint union. Therefore $\star$ is associative, the identity permutation on $\{\ast \}$ is its unit, and $$\begin{array}{c}c(\tau \star \upsilon )=c(\tau )+c(\upsilon )-1.\end{array}$$ The last identity holds because the two root cycles become one and every other cycle remains unchanged.

For ordinary permutations we use the direct sum $\sigma \oplus ︎{\sigma }^{\prime }$ defined above. Retaining $m^{+}$ and $m^{-}$ from section 2, put $${𝒟}_m:={𝔖}_{m^{+}}\times {\Re }_{m^{-}}(m\ge 1).$$ Give a decoration $\xi =(\sigma ,\tau )\in {𝒟}_m$ the bidegree $$\begin{array}{ccc}& {\mathrm{deg}}_q(\xi ):=m^{+}-c(\sigma ),{\mathrm{deg}}_t(\xi ):=c(\sigma )+c(\tau )-1.& \text{(18)}\end{array}$$ Thus ${\mathrm{deg}}_q(\xi )$ is the reflection length of the ordinary component, while ${\mathrm{deg}}_t(\xi )$ counts the ordinary cycles together with the nonroot cycles of the rooted component.

If $a$ and $b$ are not both odd, then ${(a+b)}^{+}=a^{+}+b^{+}$ and ${(a+b)}^{-}=a^{-}+b^{-}$. In this case define $$\begin{array}{ccc}& (\sigma ,\tau )\diamond ({\sigma }^{\prime },{\tau }^{\prime }):=(\sigma \oplus ︎{\sigma }^{\prime },\tau \star {\tau }^{\prime })\in {𝒟}_{a+b}.& \text{(19)}\end{array}$$

Let $A_m^{\mathrm{dec}}=𝕂[{𝒟}_m]$ for $m\ge 1$, let $A_0^{\mathrm{dec}}=𝕂1$, with $1$ a two-sided unit of bidegree $(\mathrm{0,0})$, and extend (19) bilinearly by $$\begin{array}{ccc}& A_a^{\mathrm{dec}}\diamond A_b^{\mathrm{dec}}=0\text{when }a\text{ and }b\text{ are both odd}.& \text{(20)}\end{array}$$

The product $\diamond$ makes $A^{\mathrm{dec}}={⨁}_{m\ge 0}{A}_m^{\mathrm{dec}}$ a connected bigraded associative algebra. This algebra is generally noncommutative. On every nonzero product, both degrees in (18) are additive: $${\mathrm{deg}}_q(\xi \diamond {\xi }^{\prime })={\mathrm{deg}}_q(\xi )+{\mathrm{deg}}_q({\xi }^{\prime }),{\mathrm{deg}}_t(\xi \diamond {\xi }^{\prime })={\mathrm{deg}}_t(\xi )+{\mathrm{deg}}_t({\xi }^{\prime }).$$ Indeed, on three factors containing at most one odd size, associativity follows from (17) and the associativity of $\star$. If at least two sizes are odd, both parenthesizations vanish by (20). The cycle formulas and the equalities ${(a+b)}^{+}=a^{+}+b^{+}$ and ${(a+b)}^{-}=a^{-}+b^{-}$ for every nonzero product prove degree additivity.

Let $$A^{\mathrm{even}}=\underset{m\text{ even}}{⨁}{A}_m^{\mathrm{dec}},A^{\mathrm{odd}}=\underset{m\text{ odd}}{⨁}{A}_m^{\mathrm{dec}}.$$ Then $A^{\mathrm{dec}}=A^{\mathrm{even}}\oplus ︎A^{\mathrm{odd}}$, and $A^{\mathrm{odd}}$ is a square-zero $A^{\mathrm{even}}$-bimodule.

The decorated complex.

For every nonempty composition $\alpha \models n$, set $$C_k^{t,q}(\alpha ):=\underset{\begin{array}{c}\beta ⪰\alpha \\ \ell (\beta )=k\end{array}}{⨁}{M}_{t,q}^{\beta },{\partial }_k^{t,q}:=\sum _{i=1}^{k-1}{(-1)}^{i-1}{d}_i^{t,q}.$$ Then ${({\partial }^{t,q})}^2=0$, and the differential has bidegree $(\mathrm{0,0})$. Indeed, associativity of union and $\diamond$ gives the face identities, so the standard alternating cancellation gives ${({\partial }^{t,q})}^2=0$. The cycle formulas show that the differential has bidegree $(\mathrm{0,0})$. Thus $C_{\bullet }^{t,q}(\alpha )$ is a bigraded complex of ${𝔖}_n$-modules.

The local bigraded enumerator is $$\begin{array}{c}{Q}_m(t,q):=\sum _{(\sigma ,\tau )\in {𝒟}_m}{q}^{m^{+}-c(\sigma )}{t}^{c(\sigma )+c(\tau )-1}.\end{array}$$

For every $m\ge 1$, $$\begin{array}{ccc}& Q_m(t,q)=\prod _{i=0}^{m^{+}-1}(t+iq)\prod _{j=1}^{m^{-}}(t+j).& \text{(21)}\end{array}$$ In particular, $$\begin{array}{ccc}& Q_m(t,-1)=t^{\underline{m^{+}}}\prod _{j=1}^{m^{-}}(t+j)={(t+m^{-})}^{\underline{m}}=m!\left(\genfrac{}{}{0px}{}{t+m^{-}}{m}\right).& \text{(22)}\end{array}$$ Indeed, the ordinary factor is the classical cycle enumerator (6). For a rooted permutation on $s+1$ letters, the cycle containing the root always contributes a factor $t$. Removing this factor from the usual cycle enumerator leaves ${\prod }_{j=1}^s(t+j)$. This proves (21). At $q=-1$, the factors combine to give (22).

Choosing the ordered set partition by (11) and then its block decorations shows that the bigraded Hilbert polynomial of the $k$-th chain group is $$\begin{array}{ccc}& {\mathrm{\Phi }}_{\alpha ,k}(t,q)=\sum _{\begin{array}{c}\beta ⪰\alpha \\ \ell (\beta )=k\end{array}}\left(\genfrac{}{}{0px}{}{n}{{\beta }_1,\dots ,{\beta }_k}\right)\prod _{j=1}^k{Q}_{{\beta }_j}(t,q).& \text{(23)}\end{array}$$

Decomposition and equivariant homology

A composition has at most one odd part if and only if all adjacent products in all of its coarsenings are nonzero. Indeed, if there are at least two odd parts, choose two consecutive odd parts; every intervening part is even, and absorbing those even parts into either side produces adjacent odd parts in a coarsening. Hence every iterated decoration product attached to a coarsening of such a composition is nonzero. In particular, the condition holds for every zigzag composition ${\delta }_n$.

Fix $\alpha \models n$ with at most one odd part and let $\theta =(\sigma ,\tau )\in {𝒟}_n$. For $c\in D(\alpha )$, call $c$ a factorization cut of $\theta$ if

1. $\sigma ([c^{+}])=[c^{+}]$; and

2. every nonroot cycle of $\tau$ lies entirely in $[c^{-}]$ or entirely in $\{c^{-}+1,\dots ,n^{-}\}$, while the root cycle $(\ast ,a_1,\dots ,a_u)$ has all letters at most $c^{-}$ before all letters greater than $c^{-}$.

The two sides of the cut cannot both have odd size, so these conditions are equivalent to the existence of unique decorations ${\theta }_L\in {𝒟}_c$ and ${\theta }_R\in {𝒟}_{n-c}$ such that $\theta ={\theta }_L\diamond {\theta }_R$. Set $$\begin{array}{c}{F}_{\alpha }(\theta ):=\{c\in D(\alpha ):c\text{ is a factorization cut of }\theta \},{\gamma }_{\alpha }(\theta ):={\gamma }_{F_{\alpha }(\theta )}.\end{array}$$ For the ordinary component, condition (a) is direct-sum factorization at $c^{+}$, also called shifted concatenation. Connected permutations and their factorization under this product are discussed in (Hivert et al. 2008, secs. 3.1–3.2, especially Equation (34) and Proposition 3.3). The lemma below combines this ordinary factorization with rooted factorization at the same set of cuts.

Continue to assume that $\alpha$ has at most one odd part. For $S\subseteq D(\alpha )$, set $$\begin{array}{ccc}& G_{\alpha ,S}(t,q):=\sum _{S\subseteq T\subseteq D(\alpha )}{(-1)}^{|T|-|S|}\prod _i{Q}_{{({\gamma }_T)}_i}(t,q).& \text{(25)}\end{array}$$

The cut coproduct and total-decoration fibers

The adjacent-merge differential admits a deconcatenation coproduct. A cut at a part boundary of $\alpha$ produces the corresponding prefix and suffix compositions, so the natural object is the family over all compositions and all finite label sets.

For a finite set $I$ with $|I|=n$, let ${𝔖}_I$ denote its symmetric group, and let $C_{\bullet }^{t,q}(\alpha )[I]$ denote the complex of definition 3.8, with $I$ in place of $[n]$. For the empty composition, set $C_0^{t,q}(⌀)[⌀]=𝕂$ and set all other terms equal to zero. Relabeling the blocks along a bijection of finite sets makes $$\begin{array}{c}{𝒞}_{\bullet }[I]:=\{\begin{array}{cc}{\displaystyle \underset{\alpha \models |I|}{⨁}{C}_{\bullet }^{t,q}(\alpha )[I],}& I\ne ⌀,\\ C_{\bullet }^{t,q}(⌀)[⌀],& I=⌀,\end{array}\end{array}$$ a species in bigraded chain complexes. For species $ℱ$ and $𝒢$, their Cauchy product is $$(ℱ\cdot 𝒢)[I]=\underset{I=S\bigsqcup T}{⨁}ℱ[S]\otimes 𝒢[T],$$ where the sum is over ordered decompositions; see (Aguiar and Mahajan 2010).

Let $\alpha =({\alpha }_1,\dots ,{\alpha }_r)$ and put $$c_p={\alpha }_1+\cdots +{\alpha }_p,{\alpha }_{\le p}=({\alpha }_1,\dots ,{\alpha }_p),{\alpha }_{>p}=({\alpha }_{p+1},\dots ,{\alpha }_r),$$ for $0\le p\le r$, with $c_0=0$ and with empty endpoint compositions. Let $I=S\bigsqcup T$ with $|S|=c_p$. On a basis element $$w=(B_1,\dots ,B_k;{\xi }_1,\dots ,{\xi }_k)\in C_k^{t,q}(\alpha )[I],$$ define $$\begin{array}{ccc}& {\mathrm{\Delta }}_{S,T}^p(w)=\{\begin{array}{c}(B_1,\dots ,B_j;{\xi }_1,\dots ,{\xi }_j)\otimes (B_{j+1},\dots ,B_k;{\xi }_{j+1},\dots ,{\xi }_k),\\ \text{if }S=B_1\bigsqcup \cdots \bigsqcup B_j\text{ for some }0\le j\le k,\\ 0,\text{otherwise}.\end{array}& \text{(29)}\end{array}$$ The empty prefix and suffix are interpreted as the empty bar. In the first case, the two bars lie in $C_{\bullet }^{t,q}({\alpha }_{\le p})[S]$ and $C_{\bullet }^{t,q}({\alpha }_{>p})[T]$, respectively.