Article No. ej980219Europ. J. Combinatorics(1998)19, 553–558

The Pessimistic Search and the Straightening Involution for Trees

WILLIAM Y. C. CHEN

We introduce the idea of pessimistic search on a rooted tree, and develop the straightening involutionto relate the inversion polynomial evaluated atq = −1 to the number of even rooted trees. We obtaina differential equation for the inversion polynomial of cyclic trees evaluated atq = −1, a problemproposed by Gessel, Sagan and Yeh. Some brief discussions about relevant topics are also presented.

c© 1998 Academic Press

1. INTRODUCTION

It has been noted that the inversion polynomial of rooted trees evaluated atq = −1 hasseveral combinatorial interpretations [1–6]. In this paper, we devise an involution on rootedtrees which leads to the evaluation of inversion polynomials atq = −1. This involution iscalled the straightening involution and relies on the notion of pessimistic search on a rootedtree, which appears to be more natural than the one given by Pansiot [6]. The pessimistic searchcan be described as a depth-first search guided by the minimum element in all the subtrees ofthe vertex that is being visited. The order on the vertices induced by the pessimistic searchplays a key role in the construction of the straightening involution, which gives a cancellationrule to all the rooted trees in terms of the parity of the number of inversions, and what are leftafter the cancellation turn out to be even trees (rooted trees in which every non-root vertex hasan even number of children). Moreover, the straightening involution has the property that it isautomatically valid for enriched trees (rooted trees in which the set of subtrees of any vertexis endowed with a combinatorial structure).

We mostly follow the notation and terminology in the paper of Gessel, Sagan and Yeh [3].However, as an exception, we shall stand by the commonly used method of drawing the rootof a tree at the top and use the term plane trees to refer to what Gessel, Sagan and Yeh calledordered trees, namely rooted trees in which the set of subtrees of any vertex is linearly ordered.However, we remain consistent with the notation in [3] that any rooted tree is onn verticeswhich are labelled by the numbers 1,2, . . . ,n with 1 as the root. Supposei and j are twovertices in a rooted treeT . We say thati is abovej or j is underneathi if i is on the uniquepath from j to the root 1 inT . An inversion inT is a pair of vertices(i, j ) such thati < j andj is abovei . The number of inversions ofT is denoted by inv(T). The inversion polynomialfor all rooted trees onn vertices is given by

Jn(q) =∑|T |=n

qinv(T),

where|T | denotes the number of vertices ofT .The polynomialJn(q) was determined by Mallows and Riordan [7] in terms of a generating

function. It was later discovered that this polynomial is closely related to other combinatorialenumeration problems, most notably the enumeration of connected graphs, acyclic digraphs,and Eulerian numbers [8–12]. It is worth mentioning that the Entringer numbers [8, 10, 13, 14]can also be formulated in terms of even trees.

Recently, Gessel, Sagan and Yeh [3] took a new approach to the inversion polynomial byconsidering the analogous polynomial for plane trees, where the inversion number of a plane

0195-6698/98/070553 + 06 $30.00/0 c© 1998 Academic Press

554 W. Y. C. Chen

tree is the same as that of a rooted tree, while disregarding the linear order on the set ofsubtrees of each vertex. LetQn(q) denote the inversion polynomial for the set of plane treeson n vertices (with 1 as the root, to be precise). The values ofQn(q) are determined forq = 0,1,−1. Note thatQn(0) is the number of increasing plane trees onn vertices, andQn(1) is the number of plane trees onn vertices. Moreover, Gessel, Sagan and Yeh alsoobtained the following formula forQn(−1):

Qn(−1) ={(3m− 1)!!!(3m− 2)!!!, if n = 2m+ 1 ;(3m− 2)!!!(3m− 4)!!!, if n = 2m .

(1.1)

The triple product is defined as

n!!! = n(n− 3)(n− 6) · · · ,where the product terminates with the smallest possible positive integer (either 1, 2, or 3). Thesimplicity of the above formula suggests that there exists a combinatorial interpretation, and abijective approach to this problem has been given in [15] based on the straightening involutionand a decomposition algorithm for increasing even plane trees.

The inversion polynomialCn(q) for the class of cyclic trees is also studied in [3] and theproblem of computingCn(−1) was left open. We will establish a differential equation for thegenerating functionh(x) of the sequenceC2n+1(−1). Consequently, the generating functionf (x) of the sequenceCn(−1) is determined byh(x). The solution of the differential equationfor the generating functionh(x) turns out to be analogous to the exponential integral given in[3].

The idea of pessimistic search also applies to the evaluation atx = −1 of the inversionpolynomial of binary trees on[n] = {1,2, . . . ,n} in terms of even binary trees as studied byKuznetsov, Pak and Postnikov [10].

It is worth mentioning that in recent work on a sequence of Ramanujan polynomials, Zheng[16] also considered the function of picking up the minimum element in a subtree, indicatinga potential connection with the pessimistic search. The evaluation of the aforementionedRamanujan polynomials atx = −1 was also studied by Dumont and Ramamonjisoa [17].

2. THE PESSIMISTIC SEARCH AND THE STRAIGHTENING INVOLUTION

There have been many search schemes for rooted trees, however, for the purpose of thispaper we will introduce a new search scheme which is quite different from the traditional ones.Usually, a search scheme for rooted trees is described recursively by a strategy to imposepriorities among the children of the root (the vertices immediately underneath the root). Forthe pessimistic search, the set of children of a vertex is not enough to decide the priorities,instead, the information about the minimum elements in all the subtrees is needed, and thisis the reason we introduce the term ‘pessimistic search’. Strictly speaking, the pessimisticsearch on a treeT may be recursively defined as follows. Supposer is the root ofT , and{T1, T2, . . . , Tk} is the set of subtrees of the rootr linearly ordered by the overall minimumelements contained in the subtrees, rather than the roots of the subtrees. In the pessimisticsearch one visits the rootr first, then recursively searchesT1, T2, . . . , Tk. The pessimisticorder ofT is the sequence of vertices ofT , which reflects the pessimistic search.

We are now ready to describe the straightening involution for rooted trees, which is alsovalid when a structure is endowed with the set of subtrees of each vertex. A rooted treeT iscalled increasing if it has no inversion, i.e., any path from the root to a leaf is an increasingsequence. The degree of a vertexi in T is defined as the number of its children.T is said to

Pessimistic search 555

be even if every non-root vertex has even degree. The statement for our involution is thereforeas follows.

THEOREM 2.1. The inversion polynomial Jn(q) evaluated at q= −1, is equal to thenumber of increasing even rooted trees on n vertices.

To describe the straightening involution, we need to define some terminology and notation.For a vertexi in T , we useTi to denote the subtree ofT rooted ati containing all the verticesunderneathi . A vertexi in T is said to be stable ifi is the minimum vertex inTi , and a vertexiis said to be even if it has even degree, and globally even if there are an even number of verticesunderneathi , or equivalently,|Ti | is odd.T is said to be globally even if every non-root vertexis globally even.

PROPOSITION 2.2. A rooted tree T is globally even if and only if it is even.

PROOF. First, if T is even, then it is globally even. Now we suppose that there is a non-rootvertexi in T such that the degree ofi is odd. If there is a childj of i such that|Tj | is even,then by definitionj is not globally even; otherwise for any childj of i we have|Tj | is odd,implying that|Ti | is even because the degree ofi is odd. HenceT is not globally even. 2

Despite the above equivalence of evenness and global evenness, the latter plays a subtle rolein the following involution. A non-root vertex is said to be proper if it is stable and globallyeven. A rooted treeT is said to be degenerate if every non-root vertex is proper. By theabove proposition,T is degenerate if and only if it is even and increasing. The straighteninginvolution is then a sign-reversing involution for the set of non-degenerate rooted trees onnvertices, the sign ofT is defined to be(−1)inv(T), as expected. It consists of the following twosteps:

(1) The first step of the straightening involution is where the pessimistic search comes intoplay. Leti be the first vertex in the pessimistic order that is not proper.

(2) The second step is to exchange the vertexi with a vertex j in the subtreeTi , where j isdetermined as follows. If|Ti | = 2k then relabelTi with 1′,2′, . . . , (2k)′; if |Ti | = 2k+1,then relabelTi with 0′,1′,2′, . . . , (2k)′. We make the following pairing of the verticesin Ti :

1′ ←→ 2′, 3′ ←→ 4′, . . . , (2k− 1)′ ←→ (2k)′.

The vertex(2i −1)′ is called the the companion of the vertex(2i )′, and vice versa. Nowthe vertexj is chosen as the companion of the vertexi , and they are exchanged (or morestrictly, their labels are exchanged). The resulting tree is denoted byT ′.

The following theorem asserts that the above procedure in fact yields a sign-reversing invo-lution for the set of non-degenerate rooted trees.

THEOREM 2.3. The above map T−→ T ′ is a sign-reversing involution on the set of non-degenerate rooted trees on n vertices.

PROOF. Supposei is the first vertex in the pessimistic order ofT that is not proper. Weclaim that the rooti in the subtreeTi cannot be relabeled as 0′, which is a crucial point for theinvolution. This is easy to see becausei is not proper. Supposej is the companion ofi andT ′is obtained fromT by exchanging the labelsi and j . Then inv(T) and inv(T ′) differ by one,becausei and j are adjacent in the linear order of the vertices inTi . Note thatj is not proper

556 W. Y. C. Chen

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FIGURE 1. Straightening involution.

in T ′, and it is also the first vertex in the pessimistic order ofT ′ that is not proper becausethe minimum element ofTi remains the same after the exchange ofi and j . Moreover, therelabeling ofTj in T ′ (or T ′j for short) is the same as the relabeling forTi in T . Thus,i and jremain companions to each other inT ′j , implying thatT −→ T ′ is an involution. 2

Let us give an illustration of the pessimistic search and the straightening involution inFigure 1. The pessimistic search of the first tree in Figure 1 yields the following order ofvertices:

1; 15,2,5,13,9,6,19; 7,3,8,10,18,14; 11,4; 16,12,17.

In the pairing process forT15, vertex 2 is treated 0′, and the pairing in original labels is asfollows

5←→ 6, 9←→ 13, 15←→ 19.

3. CYCLIC TREES

Note that the pessimistic search and straightening involution are also valid for rooted treesin which the set of subtrees of any vertex is endowed with a combinatorial structure, e.g.,plane trees, cyclic trees. As an application of Theorem 2.1, we derive a differential equationdetermining the generating function ofCn(−1), whereCn(q) is the inversion polynomial ofcyclic trees. By Theorem 2.3,Cn(−1) is equal to the number of even increasing cyclic trees.For short, we useCn to denoteCn(−1). We now use the standard technique of exponentialgenerating functions and the combinatorial interpretation of functional composition. Letf (x)be the generating function ofCn (n ≥ 0), whereC0 is defined to be 1, i.e.,

f (x) =∑n≥0

Cnxn

n! .

Moreover, we define the following refinements off (x) according to parity:

g(x) =∑n≥0

C2nx2n

(2n)! ,

h(x) =∑n≥0

C2n+1x2n+1

(2n+ 1)! .

Pessimistic search 557

As the number of cyclic permutations onk elements is equal to(k − 1)!, it is easy to obtainthe following recursion forCn:

Cn+1 =∑

{B1,...,Bk}C|B1|C|B2| · · ·C|Bk|(k− 1)!, (3.1)

where{B1, . . . , Bk} ranges over partitions of{1,2, . . . ,n} with each blockBi containing anodd number of elements, and|Bi | denotes the cardinality ofBi . Using the combinatorialinterpretation of functional composition of exponential generating functions, it follows from(3.1) that

f ′(x) = 1+ log1

1− h(x). (3.2)

From the above equation one sees thatf (x) is determined byh(x). To derive a differentialequation forh(x), one needs the following recursion forC2n+1:

C2n+1 =∑

{B1,...,Bk}C|B1|C|B2| · · ·C|Bk|(k− 1)!, (3.3)

where{B1, . . . , Bk} ranges over partitions of{1,2, . . . ,2n} with each blockBi containing anodd number of elements. It is important to note that the number of blocks in the partitioninvolved in (3.3) is even. As the exponential generating function for the sequence(2k− 1)! isequal to

1

2

(log

1

1− x+ log

1

1+ x

)= 1

2log

1

1− x2 .

It follows that

h′(x) = 1+ 1

2log

1

1− h(x)2. (3.4)

There do not seem to be simple formulas forCn or the generating functionsf (x), g(x) andh(x). However, we may obtain an integral formula forh(x). Clearly,h(x) is equal to thefunctional inverse of the following integral:∫

dx

1+ 12 log( 1

1−x2 ). (3.5)

Using the substitution

t = 1

2log

(1

1− x2

),

(3.5) can be expressed as ∫e2t

(t + 1)√

1− e2tdt, (3.6)

which is similar to the exponential integral in [3].

ACKNOWLEDGEMENTS

This work was performed under the auspices of the U.S. Department of Energy and theNational Science Foundation of China. I would like to thank Bruce Sagan and Jiang Zeng forstimulating discussions, and the referee for valuable suggestions.

558 W. Y. C. Chen

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(1982), 265–271.14. }}C. Poupard, Two other interpretations of the Entringer numbers,Eur. J. Combin., 18(1997), 939–943.15. }}W. Y. C. Chen, Decomposition of increasing trees, submitted.16. }}J. Zeng, A Ramanujan sequence that refines the Cayley formula for trees, preprint, 1997.17. }}D. Dumont and A. Ramamonjisoa, Grammaire de Ramanujan et Arbres de Cayley,Elec. J. Combin.,

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Received 22 December 1994 and accepted 18 February 1998

W. Y. C. CHEN

T-7, Mail Stop B284,Los Alamos National Laboratory,

Los Alamos, NM 87545,U.S.A.

Correspondence address: Research Center for Combinatorics,Nankai University,

Tianjin 300071,People’s Republic of China

E-mail: chen@t7.lanl.gov