David D. Gebhard, Bruce E. Sagan
Greene and Zaslavsky proved that the number of acyclic orientations of a graph with a unique sink is, up to sign, the linear coefficient of the chromatic polynomial. We give three new proofs of this result using pure induction, noncommutative symmetric functions, and an algorithmic bijection.
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We define the acyclic orientation polynomial of a graph to be the generating function for the sinks of its acyclic orientations. Stanley proved that the number of acyclic orientations is equal to the chromatic polynomial evaluated at $-1$ up to sign. Motivated by this link between acyclic orientations and the chromatic polynomial, we develop "acyclic orientation" analogues of theorems concerning the chromatic polynomial of Birkhoff, Whitney, and Greene-Zaslavsky. As an application, we provide a new proof for Stanley's sink theorem for chromatic symmetric functions $X_G$. This theorem gives a r…
In 1995, Richard Stanley introduced the chromatic symmetric function $X_G$ of a graph $G$ and proved that, when written in terms of the elementary symmetric functions, it reveals the number of acyclic orientations of $G$ with a given number of sinks. In this paper, we generalize this result to signed graphs, that is, to graphs whose edges are labeled with $+$ or $-$ and whose colorings and orientations can interact with their signs. Additionally, we introduce a non-homogeneous basis which detects the number of sinks and which not only gives a Stanley-type result for signed graphs but gives an …
The set of acyclic orientations of a connected graph with a given sink has a natural poset structure. We give a geometric proof of a result of Jim Propp: this poset is the disjoint union of distributive lattices.
Encoding and Enumerating Acyclic Orientations of Graphs
本文研究图的非循环定向编码与枚举，提出完全多部图的向量编码和递归构造方法，并给出若干闭公式及任意图非循环定向数目的精确上下界。
The orientation completion problem for a fixed class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class. Orientation completion problems have been studied recently for several classes of oriented graphs, yielding both polynomial time solutions and NP-completeness results. Local tournaments are a well-structured class of oriented graphs that generalize tournaments and their underlying graphs are intimately related to proper circular-arc graphs. Proper interval graphs are precisely those which can be oriented as acyclic local tourn…
Acyclic orientations and Hessenberg varieties
本文建立 Dyck 图无环定向与正则幂零 Hessenberg 簇 Tymoczko 胞腔的双射，并由此推出 Shareshian-Wachs 公式。
The main contribution of this paper is a formula for the number of acyclic orientations of a complete bipartite, $K_{n_1,n_2},$ revealing that it is equal to the poly-Bernoulli number $B_{n_1}^{(-n_2)}$ introduced in 1997 by Kaneko. We also give a simple bijective identification of acyclic orientations and lonesum matrices, confirming a 2008 result of Brewbaker, and show that the poly-Bernoulli numbers behave concavely, as well being symmetric. A second goal is to explore the behaviour of more general complete $r$-partite graphs in the space of the number of acyclic orientations. We prove that…
We show that the 4-variable generating function of certain orientation related parameters of an ordered oriented matroid is the evaluation at (x + u, y+v) of its Tutte polynomial. This evaluation contains as special cases the counting of regions in hyperplane arrangements and of acyclic orientations in graphs. Several new 2-variable expansions of the Tutte polynomial of an oriented matroid follow as corollaries. This result hold more generally for oriented matroid perspectives, with specific special cases the counting of bounded regions in hyperplane arrangements or of bipolar acyclic orientat…
Hamiltonicity of graphs of acyclic orientations and acyclic polynomials
该研究刻画了$\text{AO}$-哈密顿的多路径，提出保持$\text{AO}$-哈密顿性的粘合准则与方法，定义无环多项式并探究其性质，为$\text{AO}$-哈密顿性的归纳认证提供支撑。
Building on previous work by Cameron et al. in [3], we give a recurrence for computing the number of acyclic orientations of complete $k$-partite graphs, which can be implemented to obtain a dynamic programming algorithm running in time $n^{O(k)}$, where $n$ is the number of vertices in the graph. We prove our result by using a relationship between the number of acyclic orientations and the number of Hamiltonian paths in complete $k$-partite graphs and providing a recurrence for the latter quantity. We give a simple extension of our algorithm to the situation when we are an edge removal away f…
We present a new correspondence between acyclic orientations and coloring of a signed graph (symmetric graph). Goodall et al. introduced a bivariate chromatic polynomial $χ_G(k,l)$ that counts the number of signed colorings using colors $0,\pm1,\dots,\pm k$ along with $l-1$ symmetric colors $0_1,\dots,0_{l-1}$. We show that the evaluation of the bivariate chromatic polynomial $|χ_G(-1,2)|$ is equal to the number of acyclic orientations of the signed graph modulo the equivalence relation generated by swapping sources and sinks. We present three proofs of this fact, a proof using toric hyperplan…
We introduce polytopal cell complexes associated with partial acyclic orientations of a simple graph, which generalize acyclic orientations. Using the theory of cellular resolutions, two of these polytopal cell complexes are observed to minimally resolve certain special combinatorial polynomial ideals related to acyclic orientations. These ideals are explicitly found to be Alexander dual, which relative to comparable results in the literature, generalizes in a cleaner and more illuminating way the well-known duality between permutohedron and tree ideals. The combinatorics underlying these resu…
We extend the duality between acyclic orientations and totally cyclic orientations on planar graphs to dualities on graphs on orientable surfaces by introducing boundary acyclic orientations and totally bi-walkable orientations. In addition, we provide a reciprocity theorem connecting local tensions and boundary acyclic orientations. Furthermore, we define the balanced flow polynomial which is connected with tension polynomial by duality and with totally bi-walkable orientations by reciprocity.
A unique sink orientation (USO) is an orientation of the $n$-dimensional cube graph ($n$-cube) such that every face (subcube) has a unique sink. The number of unique sink orientations is $n^{Θ(2^n)}$. If a cube orientation is not a USO, it contains a pseudo unique sink orientation (PUSO): an orientation of some subcube such that every proper face of it has a unique sink, but the subcube itself hasn't. In this paper, we characterize and count PUSOs of the $n$-cube. We show that PUSOs have a much more rigid structure than USOs and that their number is between $2^{Ω(2^{n-\log n})}$ and $2^{O(2^n)…
e-basis Coefficients of Chromatic Symmetric Functions
本文推广 Stanley 关于色对称函数 e-基系数与无环定向汇点序列的结果，证明顶点加权版本，并提出可解释更多 e-系数和的更强猜想。
The chromatic polynomial $P(G,x)$ of a graph $G$ of order $n$ can be expressed as $\sum\limits_{i=1}^n(-1)^{n-i}a_{i}x^i$, where $a_i$ is interpreted as the number of broken-cycle free spanning subgraphs of $G$ with exactly $i$ components. The parameter $ε(G)=\sum\limits_{i=1}^n (n-i)a_i/\sum\limits_{i=1}^n a_i$ is the mean size of a broken-cycle-free spanning subgraph of $G$. In this article, we confirm and strengthen a conjecture proposed by Lundow and Markström in 2006 that $ε(T_n)< ε(G)<ε(K_n)$ holds for any connected graph $G$ of order $n$ which is neither the complete graph $K_n$ nor a t…
We study graphs which admit an acyclic orientation that contains an out-branching and in-branching which are arc-disjoint (such an orientation is called {\bf good}). A {\bf 2T-graph} is a graph whose edge set can be decomposed into two edge-disjoint spanning trees. Clearly a graph has a good orientation if and only if it contains a spanning 2T-graph with a good orientation, implying that 2T-graphs play a central role. Vertex-minimal 2T-graphs with at least two vertices, also known as {\bf generic circuits}, play an important role in rigidity theory for graphs. It was shown in \cite{bangGOpaper…
Unique-sink orientations (USOs) are an abstract class of orientations of the n-cube graph. We consider some classes of USOs that are of interest in connection with the linear complementarity problem. We summarise old and show new lower and upper bounds on the sizes of some such classes. Furthermore, we provide a characterisation of K-matrices in terms of their corresponding USOs.
A sink-free orientation of a finite undirected graph is a choice of orientation for each edge such that every vertex has out-degree at least 1. Bubley and Dyer (1997) use Markov Chain Monte Carlo to sample approximately from the uniform distribution on sink-free orientations in time O(m^3 log (1/epsilon)), where m is the number of edges and epsilon the degree of approximation. Huber (1998) uses coupling from the past to obtain an exact sample in time O(m^4). We present a simple randomized algorithm inspired by Wilson's cycle popping method which obtains an exact sample in mean time at most O(n…
We examine a dynamic model for the disruption of information flow in hierarchical social networks by considering the vertex-pursuit game Seepage played in directed acyclic graphs (DAGs). In Seepage, agents attempt to block the movement of an intruder who moves downward from the source node to a sink. The minimum number of such agents required to block the intruder is called the green number. We propose a generalized stochastic model for DAGs with given expected total degree sequence. Seepage and the green number is analyzed in stochastic DAGs in both the cases of a regular and power law degree…
The Matoušek LP-type problems were used by Matoušek to show that the Sharir-Welzl algorithm may require at least subexponential time. Later, Gärtner translated this result into the language of Unique Sink Orientations (USOs) and introduced the Matoušek USOs, the USOs equivalent to Matoušek's LP-type problems. He further showed that the Random Facet algorithm only requires quadratic time on the realizable subset of the Matoušek USOs, but without characterizing this subset. In this paper, we deliver this missing characterization and also provide concrete realizations for all realizable Matoušek …
Signotopes Induce Unique Sink Orientations on Grids
研究通过推广线性函数诱导的USO特性，探讨了r维网格和(r+1)符号otope上的USO结构特征。
We call a topological ordering of a weighted directed acyclic graph non-negative if the sum of weights on the vertices in any prefix of the ordering is non-negative. We investigate two processes for constructing non-negative topological orderings of weighted directed acyclic graphs. The first process is called a mark sequence and the second is a generalization called a mark-unmark sequence. We answer a question of Erickson by showing that every non-negative topological ordering that can be realized by a mark-unmark sequence can also be realized by a mark sequence. We also investigate the quest…
In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. Firstly, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage-Squire-West construction with a recent algorithm for generating elimination trees of chordal graphs. Secondly, we consider quotients of lattices of acyclic orientations of ch…
Acyclic orientations of mixed graphs
研究无环混合图的定向完成问题，证明若干判定问题的复杂性，并给出一个多项式算法，提出开放问题。
Given an undirected graph $G=(V,E)$, and a designated vertex $q\in V$, the notion of a $G$-parking function (with respect to $q$) was independently developed and studied by various authors, and has recently gained renewed attention. This notion generalizes the classical notion of a parking function associated with the complete graph. In this work, we study properties of {\em maximum} $G$-parking functions and provide a new bijection between them and the set of spanning trees of $G$ with no broken circuit. As a case study, we specialize some of our results to the graph corresponding to the disc…
Note about the complexity of the acyclic orientation with parity constraint problem
本文研究了具有奇偶约束的无环定向问题的复杂性，证明了在某些图中该问题为NP难。
Let D be an acyclic orientation of a simple graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let d(D) denote the number of dependent arcs in D. Define m and M to be the minimum and the maximum number of d(D) over all acyclic orientations D of G. We call G fully orientable if G has an acyclic orientation with exactly k dependent arcs for every k satisfying m <= k <= M. In this paper, we prove that the square of a cycle C_n of length n is fully orientable except n=6.
A matroidal criterion for flow polytopes to be order polytopes
研究有向无环图流多胞体与序多胞体的关系，核心方法是收缩闲置边，主要贡献为证明\(\mathcal{F}(G)\)与序多胞体单模等价当且仅当\(\widetilde G\)是\(st -\)平面图，以及在特定条件下给出图是\(st -\)平面图的充要条件。
Some polynomial classes for the acyclic orientation with parity constraint problem
本文研究了无环定向问题中的奇偶约束，提出多项式类以确定无环T-奇定向的存在条件，并通过构造性证明展示了其在特定图结构中的可解性。
A unique sink orientation (USO) is an orientation of the hypercube graph with the property that every face has a unique sink. A number of well-studied problems reduce in strongly polynomial time to finding the global sink of a USO; most notably, linear programming (LP) and the P-matrix linear complementarity problem (P-LCP). The former is not known to have a strongly polynomial-time algorithm, while the latter is not known to even have a polynomial-time algorithm, motivating the problem to find the global sink of a USO. Although, every known class of geometric USOs, arising from a concrete pro…
Given an acyclic oriented graph $\vec{H}$ and a graph $G$, we write $G \to \vec{H}$ if every orientation of $G$ has an oriented copy of $\vec{H}$. We define $\vec{R}(\vec{H})$ as the smallest number $n$ such that there exists a graph $G$ satisfying $G \to \vec{H}$. Denoting by $R(H)$ the classical Ramsey number of a graph $H$, we show that $\vec{R}(\vec{H}) \leq 2R(H)^{c \log^2 h}$ for every acyclic oriented graph $\vec{H}$ with $h$ vertices, where $H$ is its underlying undirected graph. We also study the threshold function for the event $\{G(n,p) \to \vec{H}\}$ in the binomial random graph $G…
Recall that an excedance of a permutation $π$ is any position $i$ such that $π_i > i$. Inspired by the work of Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it gets sorted by a certain sequence of toppling moves. One of our main results is that the number of toppleable permutations on $n$ letters is the same as those for which excedances happen exactly at $\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Additionally, we show that the above is also the number of acyclic orientations with unique sink (AUSOs) of the complete bipa…
The orientation completion problem for a class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class by orienting the unoriented edges of the partially oriented graph. Orientation completion problems have been studied recently for several classes of oriented graphs, yielding both polynomial time solutions as well as NP-completeness results. Local tournaments are a well-structured class of oriented graphs that generalize tournaments and their underlying graphs are intimately related to proper circular-arc graphs. According to Skrien,…
Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements
本文通过唯一汇定向和雨伞排列提出了新的方法，证明了 α-汉 Sandwich 定理并推广到定向子模领域。
Let c(G) be the smallest number of edges we have to test in order to determine an unknown acyclic orientation of the given graph G in the worst case. For example, if G is the complete graph on n vertices, then c(G) is the smallest number of comparisons needed to sort n numbers. We prove that c(G)\le (1/4+o(1))n^2 for any graph G on n vertices, answering in the affirmative a question of Aigner, Triesch, and Tuza [Discrete Mathematics, 144 (1995) 3-10]. Also, we show that, for every e>0, it is NP-hard to approximate the parameter c(G) within a multiplicative factor 74/73-e.
The distribution of the length of the longest path in random acyclic orientations of a complete bipartite graph
本文研究了完全二部图随机无环定向中最长路径长度的分布，给出了概率生成函数，并利用解析组合学证明了在两部分大小相等趋于无穷时分布渐近服从高斯分布，回答了 Peter J. Cameron 的问题。
Non-Promise Version of Unique Sink Orientations
本文研究唯一汇点定向的非承诺版本，将承诺版本的算法与性质适配到非承诺情形，证明其属于 PLS^dt，并将可验证违反从 2 个顶点推广到 4 个顶点并给出完整分类。
On Phases of Unique Sink Orientations
本文研究 USO 中相位的结构，给出 O(3^n) 的相位计算算法，并证明电路简洁编码的无环 USO 中边同相位判定问题是 PSPACE 完全的。
In this note we study the complexity of some generalizations of the notion of $st$-numbering. Suppose that given some functions $f$ and $g$, we want to order the vertices of a graph such that every vertex $v$ is preceded by at least $f(v)$ of its neighbors and succeeded by at least $g(v)$ of its neighbors. We prove that this problem is solvable in polynomial time if $fg\equiv 0$, but it becomes NP-complete for $f\equiv g \equiv 2$. This answers a question of the first author posed in 2009.
Sortable Elements for Quivers with Cycles
本文将Coxeter群中c-排序元素的概念推广到任意定向的箭图Ω，定义了Ω-排序元素，并证明了其关键性质（如通过约化到无环情形），为推广到非无环簇代数奠定基础。
Unique Sink Orientations (USOs) of cubes can be used to capture the combinatorial structure of many essential algebraic and geometric problems. For various structural and algorithmic questions, including enumeration of USOs and algorithm analysis, it is crucial to have systematic constructions of USOs. While some construction methods for USOs already exist, each one of them has some significant downside. Most of the construction methods have limited expressivity -- USOs with some desired properties cannot be constructed. In contrast, the phase flips of Schurr can construct all USOs, but the op…
Let P a polytope and let G(P) be the graph of P. Following Gil Kalai, we say that an acyclic orientation O of G(P) is good if, for every non-empty face F of P, the induced graph G(F) has exactly one sink. Gil Kalai gave a simple way to tell a simple polytope from the good orientations of its graph. This article is a broader study of ``good orientations'' (of the graphs) on matroid polytopes.
In an earlier paper, the first two authors defined orientations on hypergraphs. Using this definition we provide an explicit bijection between acyclic orientations in hypergraphs and faces of hypergraphic polytopes. This allows us to obtain a geometric interpretation of the coefficients of the antipode map in a Hopf algebra of hypergraphs. This interpretation differs from similar ones for a different Hopf structure on hypergraphs provided recently by Aguiar and Ardila. Furthermore, making use of the tools and definitions developed here regarding orientations of hypergraphs we provide a charact…
Unique Sink Orientations (USOs) are an appealing abstraction of several major optimization problems of applied mathematics such as for instance Linear Programming (LP), Markov Decision Processes (MDPs) or 2-player Turn Based Stochastic Games (2TBSGs). A polynomial time algorithm to find the sink of a USO would translate into a strongly polynomial time algorithm to solve the aforementioned problems---a major quest for all three cases. In addition, we may translate MDPs and 2TBSGs into the problem of finding the sink of an acyclic USO of a cube, which can be done using the well-known Policy Iter…
Let $G$ be a simple graph and let $\mathcal{L}(G)$ be the free partially commutative Lie algebra associated to $G$. In this paper, using heaps of pieces, we prove an expression for the generalized $\textbf k$-chromatic polynomial of $G$ in terms of dimensions of the grade spaces of $\mathcal{L}(G)$. This will give us a new interpretation for the chromatic polynomials in terms of multilinear heaps and Lyndon length. The classical results of Stanley, and Greene and Zaslavsky regarding the acyclic orientations of $G$ are obtained as corollaries. A heap with a unique minimal piece is said to be a …
The cyclic and dihedral groups can be made to act on the set Acyc(Y) of acyclic orientations of an undirected graph Y, and this gives rise to the equivalence relations ~kappa and ~delta, respectively. These two actions and their corresponding equivalence classes are closely related to combinatorial problems arising in the context of Coxeter groups, sequential dynamical systems, the chip-firing game, and representations of quivers. In this paper we construct the graphs C(Y) and D(Y) with vertex sets Acyc(Y) and whose connected components encode the equivalence classes. The number of connected c…
For an acyclic directed graph with multiple sources and multiple sinks, we prove that one can choose the Merger's paths between the sources and the sinks such that the number of mergings between these paths is upper bounded by a constant depending only on the min-cuts between the sources and the sinks, regardless of the size and topology of the graph. We also give bounds on the minimum number of mergings between these paths, and discuss how it depends on the min-cuts.
Suppose that D is an acyclic orientation of a graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let m and M denote the minimum and the maximum of the number of dependent arcs over all acyclic orientations of G. We call G fully orientable if G has an acyclic orientation with exactly d dependent arcs for every d satisfying m <= d <= M. A graph G is called chordal if every cycle in G of length at least four has a chord. We show that all chordal graphs are fully orientable.
In this paper we study the equivalence relation on the set of acyclic orientations of a graph Y that arises through source-to-sink conversions. This source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a Coxeter group. We give a direct proof of a recursion for the number of equivalence classes of this relation for an arbitrary graph Y using edge deletion and edge contraction of non-bridge edges. We conclude by showing how this result may also be obtained through an evaluation of the Tutte polynomial as T(Y,1,0), and we provide bijections to two other classes of acyclic or…
A \emph{Golomb ruler} is a sequence of distinct integers (the \emph{markings} of the ruler) whose pairwise differences are distinct. Golomb rulers can be traced back to additive number theory in the 1930s and have attracted recent research activities on existence problems, such as the search for \emph{optimal} Golomb rulers (those of minimal length given a fixed number of markings). Our goal is to enumerate Golomb rulers in a systematic way: we study [g_m(t) := # {\x \in \Z^{m+1} : \, 0 = x_0 < x_1 < ... < x_{m-1} < x_m = t, \text{all} x_j - x_k \text{distinct}},] the number of Golomb rulers w…
Given an undirected graph $G = (V, E)$, and a vertex $r\in V$, an $r$-acyclic orientation of $G$ is an orientation $OE$ of the edges of $G$ such that the digraph $OG = (V, OE)$ is acyclic and $r$ is the unique vertex with indegree equal to 0. For $w\in \mathbb{R}^E_+$, $k(G, w)$ is the value of the $w$-maximum packing of $r$-arborescences for all $r\in V$ and all $r$-acyclic orientations $OE$ of $G$. In this case, the Broadcast Routing (in Computers Networks) Problem (BRP) is to compute $k(G, w)$, by finding an optimal $r$ and an optimal $r$-acyclic orientation. BRP is a mathematical formulati…
We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in terms of elementary symmetric functions. Analysis of some of the combinatorial implications of this expansion leads to three bijections involving acyclic orientations.
If $G$ is a graph and $\vec H$ is an oriented graph, we write $G\to \vec H$ to say that every orientation of the edges of $G$ contains $\vec H$ as a subdigraph. We consider the case in which $G=G(n,p)$, the binomial random graph. We determine the threshold $p_{\vec H}=p_{\vec H}(n)$ for the property $G(n,p)\to \vec H$ for the cases in which $\vec H$ is an acyclic orientation of a complete graph or of a cycle.
Arc weighted acyclic orientations and variations of degeneracy of graphs
本文推广无环定向至弧赋权定向，定义新的$ST^{(2)}$-$f$-退化度变体，否定Richter关于3-连通非完全平面图截断度可选性的问题，证明此类图及真子图闭族中对应图的相关可选性。
Shift Graphs, Chromatic Number and Acyclic One-Path Orientations
本文利用移位图及其诱导子图，改进了完全二分图情形下无F诱导子图的色数上界，并证明移位图无AOP性质，同时构造了高奇围长但无AOP性质的图。
For an oriented graph $G$, let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of $G$. Let $f(n)$ be the smallest integer such that every oriented graph $G$ with chromatic number larger than $f(n)$ has $f(G) > n$. Let $g(n)$ be the smallest integer such that every tournament $G$ with more than $g(n)$ vertices has $f(G) > n$. It is straightforward that $Ω(n) \le g(n) \le f(n) \le n^2$. This paper provides the first nontrivial lower and upper bounds for $g(n)$. In particular, it is proved that $\frac{1}{4}n^{8/7} \le g(n) \le n^2-(2-\frac{1}{\sqrt{2}})n+2$. It is also shown tha…
The Merino-Welsh conjecture asserts that the number of spanning trees of a graph is no greater than the maximum of the numbers of totally cyclic orientations and acyclic orientations of that graph. We prove this conjecture for the class of series-parallel graphs.
Framing Triangulations and Framing Posets of Planar DAGs with Nontrivial Netflow Vectors
研究平面DAG的流量多面体在特定平面条件下，通过框架数据构建单模三角化，并定义其最大单纯形的格点结构，扩展了Tamari格与弱序的理论。
This is the first one of a series of papers on association of orientations, lattice polytopes, and abelian group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the…
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