Discrete Mathematics 241 (2001) 235–240www.elsevier.com/locate/disc

A bijective proof of the equation linking the Schr'odernumbers, large and small

Emeric DeutschDepartment of Mathematics, Polytechnic University, 333 Jay Street, Brooklyn,

NY 11201, USA

Abstract

A bijection is de*ned between the set of all bushes with n + 1 leaves and the set of allSchr'oder paths of semilength n. This shows bijectively the relation Rn = 2Sn between the largeand small Schr'oder numbers. c© 2001 Elsevier Science B.V. All rights reserved.

MSC: 05A15

Keywords: Small and large Schr'oder numbers

1. Introduction

The small Schr&oder numbers Sn; for n¿ 0; occur in various enumeration problems.We mention only three of them:

(i) Sn is the number of ordered trees with no node of outdegree one and with n+ 1leaves;

(ii) Sn is the number of dissections of a convex (n+ 2)-gon;(iii) Sn is the number of generalized bracketings of a string of n + 1 letters, as *rst

considered by Schr'oder.

The equivalence of these three de*nitions can be easily shown with the aid of well-knownbijections (see, for example, [8,9]). From any of the three de*nitions one can *nd bystraightforward counting that S0 = 1; S1 = 1; S2 = 3; S3 = 11; S4 = 45.

We would like to point out Stanley’s captivating exposition [8] on these numbers,including historical facts going back to the 2nd century B.C.

E-mail address: deutsch@magnus.poly.edu (E. Deutsch).

0012-365X/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.PII: S0012 -365X(01)00122 -4

236 E. Deutsch /Discrete Mathematics 241 (2001) 235–240

The large Schr&oder numbers Rn; for n¿ 0; also occur in various enumeration prob-lems. Again we mention only three of them:

(I) Rn is the number of lattice paths in the plane from (0; 0) to (2n; 0) with steps(1; 1); (2; 0); and (1;−1); that never go below the horizontal axis (these pathswill be called Schr&oder paths of semilength n);

(II) Rn is the number of lattice paths in the plane from (0; 0) to (2n; 0) with steps(1; 1); and (1;−1); that never go below the horizontal axis (i.e. Dyck paths) witheach peak colored either black or white;

(III) Rn is the number of parallelogram polyominoes of perimeter 2n + 2 with eachcolumn colored either black or white.

The equivalence of de*nitions (I) and (II) follows from a very simple bijection:color the peaks white and then replace each (2; 0) step by a peak (i.e. a pair of steps(1; 1); (1;−1)); colored black. The equivalence of de*nitions (II) and (III) follows atonce from a well-known bijection between Dyck paths and parallelogram polyominoes(see, for example, [2, p. 182]). From any of the three de*nitions one can *nd bystraightforward counting that R0 = 1; R1 = 2; R2 = 6; R3 = 22; R4 = 90.

Other occurrences of the small and large Schr'oder numbers are listed in [9]. Somepertinent references are [1,4–6].

It is well known that Rn = 2Sn for n¿ 1 (see, for example, [9, pp. 178, 219, 256]).Its simple, basically known analytic proof is given in the next section. Recently, Shapiroand Sulanke [7] and Sulanke [10] have given a bijective proof of this equality. Thepurpose of this note is to give another bijective proof (Section 3).

2. A generating function proof

In this section we prove Rn = 2Sn for n¿ 1 using generating functions. This proof,due to its straightforward nature, is certainly not original and it is presented to giveperspective to the bijective proof of Section 3.

An ordered tree in which no node, the root excepted, has outdegree equal to 1is called a bush (see [3]). A bush having root of degree 1 will be called a tallbush (it is also called a planted bush), while a bush having root of degree at least2 or equal to 0 (in the case of the empty bush, consisting only of the root) willbe called a short bush. Thus, the small Schr'oder number Sn gives the number ofshort bushes with n + 1 leaves. Let S(z) be the generating function of the smallSchr'oder numbers, i.e. S(z) = S0 + S1z + S2z2 + · · · . Then the generating functionT (z) for the number of short bushes according to number of leaves is given byT (z) = zS(z). Since a short bush with root of degree k can be identi*ed with ak-tuple of short bushes, with the preservation of the total number of leaves, it fol-lows that the generating function of the short bushes with root of degree k is Tk .Consequently,

T − z=T 2 + T 3 + T 4 + · · · : (1)

E. Deutsch /Discrete Mathematics 241 (2001) 235–240 237

The term z in the left-hand side of the above equation relates to the empty bush, con-sisting only of the root, which has to be viewed as a leaf. Taking into account thatT = zS; from (1) we obtain

2zS2 − (1 + z)S + 1 = 0: (2)

Let R(z) be the generating function of the large Schr'oder numbers, i.e. R(z) =R0 +R1z+R2z2+· · · . Making use of de*nition (I), decomposing the set of all Schr'oder pathsinto the empty path, the set of paths that start with a (2; 0) step, and the set of pathsthat start with a (1; 1) step, and applying to the latter the *rst return decomposition, 1

one can easily derive that

R= 1 + zR+ zR2: (3)

An elementary computation shows that the combinatorially meaningful solutions Sand R in (2) and (3), respectively, satisfy

R= 2S − 1;

from which Rn = 2Sn for n¿ 1.

3. A bijective proof

In this section we give a bijective proof of the relation Rn = 2Sn for n¿ 1. Let Sn

denote the set of short bushes with n+1 leaves and let Bn denote the set of all busheswith n+ 1 leaves. Clearly,

|Bn|= 2|Sn|= 2Sn;

since the short and the tall bushes with the same number of leaves are in an obviousbijection.

Let Rn denote the set of all Schr'oder paths of semilength n. By de*nition,

|Rn|=Rn:Before we de*ne a bijection between the sets Bn and Rn we introduce the followingterminology. If ∈Bn; then the following edges of will be called excess edges:

(a) every edge emanating from the root with the exception of the leftmost edge;(b) every edge emanating from a nonroot node with the exception of the two extreme

edges.

In Fig. 1a the excess edges are shown with dashed lines. The only bushes that haveno excess edges are the planted full binary trees.

1 This means the factorization of the path by cutting it at its *rst return to the horizontal axis.

238 E. Deutsch /Discrete Mathematics 241 (2001) 235–240

Fig. 1.

Now we de*ne a bijection :Bn → Rn. If ∈Bn; then () is obtained in thefollowing manner: traverse from the left in a preorder manner so that when eachedge is encountered for the *rst time

(i) do nothing for the *rst edge;(ii) draw a (1; 1) step for the leftmost edge emanating from a nonroot node;(iii) draw a (2; 0) step for an excess edge;(iv) draw a (1;−1) step for the rightmost edge emanating from a nonroot node.

It is easy to see that the semilength of the Schr'oder path () is equal to the numberof edges less the number of internal nodes. This, in turn is equal to one less thanthe number of leaves. Consequently, ()∈Rn. Conversely, given a Schr'oder path�∈Rn; we draw a tree in a preorder manner by the following rule: we start withan edge and, then, traversing � from left to right, for each up step (i.e., (1; 1)) wedraw a leftmost edge, for each horizontal step we draw an excess edge and for eachdown step (i.e., (1;−1)) we draw a rightmost edge emanating from the appropriatenode.

Fig. 1 illustrates the bijection. The edges of the tree are numbered in a preordermanner, starting with 0, while the steps of the Schr'oder path are numbered in se-quence, starting with 1. By de*nition, these steps correspond to the edges having

E. Deutsch /Discrete Mathematics 241 (2001) 235–240 239

Fig. 2.

the same identi*cation number. In Fig. 2 we give the correspondences between allbushes with 1, 2, and 3 leaves and all Schr'oder paths of semilength 0, 1, and 2,respectively.

Acknowledgements

The author expresses his appreciation to the referees for their careful reading of themanuscript and helpful suggestions. Many thanks to Peter Rost for the drawings.

References

[1] J. Bonin, L. Shapiro, R. Simion, Some q-analogues of the Schr'oder number arising from combinatorialstatistics on lattice paths, J. Statist. Plann. Inference 34 (1993) 35–55.

[2] M.P. Delest, G. Viennot, Algebraic languages and polyominoes enumeration, Theoret. Comput. Sci. 34(1984) 169–206.

[3] R. Donaghey, L.W. Shapiro, Motzkin numbers, J. Combin. Theory Ser. A 23 (1977) 291–301.[4] D.G. Rogers, The enumeration of a family of ladder graphs, Part II: Schr'oder and superconnective

relations, Quart. J. Math. Oxford (2) 31 (1980) 491–506.[5] D.G. Rogers, L.W. Shapiro, Some correspondences involving the Schr'oder numbers and relations.

Combinatorial Mathematics, Proceedings of the International Conference on Combinatorial TheoryCanberra, August 16–27, 1977, Lecture Notes in Mathematics, Vol. 686, Springer, Berlin, 1978,pp. 267–274.

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[6] D.G. Rogers, L.W. Shapiro, Deques, trees and lattice paths, in: K.L. McAvaney (Ed.), CombinatorialMathematics VIII, Proceedings of the Eighth Australian Conference on Combinatorial Mathematics,Geelong, Australia, August 25–29, 1980, Lecture Notes in Mathematics, Vol. 884, Springer, Berlin,1981, pp. 293–303.

[7] L.W. Shapiro, R.A. Sulanke, Bijections for the Schr'oder numbers, Mathematics Magazine 73 (2000)369–376.

[8] R.P. Stanley, Hipparchus, Plutarch, Schr'oder and Hough, Amer. Math. Monthly 104 (4) (1997)344–350.

[9] R.P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999.[10] R.A. Sulanke, The Narayana distribution, J. Statist. Plann. Inference, to appear.