Bell numbers and k-trees

Bell numbers and k-trees

DISCRETE MATHEMATICS ELSEVIER Discrete Mathematics 156 (1996) 247-252 Communication Bell numbers and k-trees W i n s t o n Yang* Department of Math...

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DISCRETE MATHEMATICS ELSEVIER

Discrete Mathematics 156 (1996) 247-252

Communication

Bell numbers and k-trees W i n s t o n Yang* Department of Mathematics, California Institute of Technology, Box 927, Pasadena, CA 91125, USA Received 30 January 1996

Abstract We show that the number of partitions of { 1, . . . , n} with the restriction that each integer i is

not in the same part as i - 1 is the Bell number B,_ 1. We generalize this result to partitions of the vertices of k-trees.

1. Introduction A partition of a set S is a set of nonempty, disjoint sets (called 'parts') whose union is S. A partition of the empty set is defined to be the empty set. The Bell number B, is the number of partitions of a set of n elements. The first few Bell numbers, starting with Bo, are 1, 1, 2, 5, 15, 52, 233. They are named after Eric Temple Bell (1883-1960). Closely related is the Stirling number of the second kind S(n, t), which is the number of partitions of a set of n elements into t parts; S(0, 0) is defined to be 1, and S(n, t) is defined to be 0 for values of n and t for which the previous two definitions are not applicable. Stirling numbers (of the first and second kind) are named after James Stirling (1692-1770). We have that B, = ~ = 1S(n, t). This paper was motivated by counting 'columns,' objects considered by Ronald C. Read in a 1962 article 'Contributions to the Cell Growth Problem' [4]. We had to count the number of partitions of the set {1. . . . ,n} with the restriction that each integer i is not in the same part as i - 1. We call such partitions 'restricted.' We call partitions as defined above 'unrestricted.' Theorem 1. The number of restricted partitions of { 1.... , n} is B._ 1.

* Email: [email protected]. 0012-365X/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S00 1 2 - 3 6 5 X ( 9 6 ) 0 0 0 3 4 - 9

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We give an example and two proofs. We represent a given partition of the ordered set {1, ... ,n} by the ordered set {p(1), ... ,p(n)}, where p(i) is the part t h a t / i s in. We use the following two rules. 1./9(1) = 1. 2. p(i) ~< 1 + max(p(1) . . . . . p(i - 1)). Rule 2 ensures that i f p a r t j is used, then parts 1, ... ,j - 1 have already been used. For example, one partition of {1, 2, 3, 4, 5} is {1, 2, 2, 1, 3}. There are three parts of the partition: 1 and 4 are in part 1; 2 and 3 are in part 2; 5 is in part 3. Example 1. The left side of the table below lists the restricted partitions of {1 . . . . ,5}. According to Theorem 1, the number of partitions is B s - 1 = B4 = 15. The right side of the table lists the unrestricted partitions of {1 . . . . ,4}. The table will be used to motivate proof 2.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1

2

3

4

5

1

2

3

4

1 1 1 1 1 1 ! 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 1 1 1 1 3 3 3 3 3 3 3 3 3 3

2 2 3 3 3 1 1 1 2 2 2 4 4 4 4

1 3 1 2 4 2 3 4 1 3 4 I 2 3 5

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 2 2 2 2 2 2 2 2 2 2

1 1 2 2 2 1 1 1 2 2 2 3 3 3 3

1 2 I 2 3 1 2 3 1 2 3 1 2 3 4

We now give two proofs of Theorem 1. Proof 1. Define a break of a partition {p(1) . . . . ,p(n)} as an integer i such that p(i) ~ p(i + 1). For example, the partition { 1, 1, 1, 2, 2, 3, 1, 3, 2} has 5 breaks. Let A(n, b) be the number of partitions of { 1. . . . . n} that have b breaks. We want to show that A(n, n - 1) = B,_ 1. First, we note that A(n, b) = A(b + 1, b) ("ba). We prove the above statement. Let {p(1) . . . . . p(n)} be a partition counted by A(n, b). Let il . . . . . ib be all the breaks in {p(1) . . . . . p(n)}; then the integers at the positions

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between two breaks i,, and i,, + 1 are equal to the integer at im+ 1. Delete these repeated integers; the result is {p(il), ..., p(ib), p(n)}, which is a partition counted by A(b + 1, b). We can reverse this process, and we can show that this map is one-to-one. Therefore the n u m b e r of partitions of {1, . . . , n} with b breaks in fixed positions is A(b, b + 1). The number of ways that b breaks can occur in a partition of {1 . . . . ,n} is (,~1). Therefore A(n, b) = A(b + 1, b) ("~ 1). F r o m the definition of A(n, b), and B,, we have that n-1

n-1

b=o

b=o

b

"

It is well known that the Bell numbers satisfy the recurrence n--1

B.=

Bb

(.1) b

'

(see e.g. [6]). A(1, 0 ) = 1 = Bo. By induction, we have that A(b + 1, b)= Bb. By a change of variable, we have A(n, n - 1) = B,_ 1. This completes the proof. [] Proof 2. In Example 1, we constructed the restricted partitions {p(1). . . . . p(5)} of {1 . . . . ,5} by considering the possible values of each p(i), given p(1) . . . . , p ( i - 1). Example 1 suggests a bijection from the restricted partitions of {1, . . . , n } to the unrestricted partitions of {1, . . . , n - 1}. We construct a bijection. Let {p(1), . . . , p(n)} be a restricted partition of { 1, ..., n}. Let look (i) be the ordered set of integers that p(i + 1) can be, arranged from least to greatest. ('look' is short for 'lookahead set', a term in computer science.) Let pos(i) ('pos') be the position ofp(i + 1) in look(i). That is, if p(i + 1) is the k-th element of look(i), then pos(i)= k. The following f u n c t i o n f i s a bijection from restricted partitions of {1 . . . . , n} to unrestricted partitions of { 1, . . . , n - 1}. We omit the details. f({p(1), ..,,p(n)}) = {pos(1), . . . , pos(n - 1)}. F o r the case n = 1, {p(1)} = {1}. We definef({p(1)}) to be the empty set.

2. Partitions of vertices of graphs A partition of a set can be thought of as a 'coloring' of the elements in the set; elements have the same colors iff they are in the same part. A proper coloring of a graph is a coloring in which adjacent vertices have different colors. We say that two colorings are isomorphic if one can be obtained from the other by a permutation of the colors. With this terminology, the problem of counting restricted partitions of {1, . . . , n} is a special case of the following problem: F o r a graph G of a finite number of vertices,

250

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what is the number of non-isomorphic proper colorings of G? Let nipc(G) be the number of such colorings of G ('nipc' stands for non-isomorphic proper colorings). A k-tree of n vertices is defined recursively as follows [2]. If n = k, then the k-tree is the complete graph Kk. If n > k, then the k-tree is formed from a k-tree of n - 1 vertices by adding a vertex, and joining it with k edges to k vertices of a complete graph Kk in the existing graph. Example 2. Fig. 1 is a 1-tree of 5 vertices, and Fig. 2 is a 2-tree of 5 vertices.

Theorem 2. Let G, be a k-tree of n vertices. Then nipc(G,)

= B n _ k.

Proof. Let nipc(G,, r) be the number of non-isomorphic proper colorings of G, with r colors. We have that n

nipc(G.) = Y. nipc(G., r). r=l

We find a recursive formula for nipc(G., r). We can construct G. from a k-tree Gn- i of n - 1 vertices by adding a vertex, and joining it with k vertices of a complete graph Kk in G._ 1. There are two cases. Case 1. The new vertex has a color already in G._ 1. The k vertices to which the new vertex is joined have different colors, because they are vertices of a complete graph Kk. Therefore the new vertex can have r - k colors. Therefore we can form ( r - k) nipc(G,_ 1, r) non-isomorphic proper colorings of G,. Case 2. The new vertex does not have a color already in G._ 1. Therefore we can form nipc(G._ 1, r - 1) non-isomorphic proper colorings of G,. From the two cases, we have that nipc(G,, r ) = ( r - k ) nipc(G._l, r ) + nipc (G,-1, r - 1). The Stirling numbers satisfy the recurrence S(n, t ) = t S ( n - 1, t) + S ( n - 1, t - 1) (see e.g. [5]). nipc(Kk+l, k + 1 ) = 1 =S(1, 1). By induction,

Fig. 1.

Fig. 2.

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Fig. 3. nipc(G,, r) = S(n - k, r - k) for r = k + 1, ..., n. Therefore n

nipc(G.) =

nipc(G., r) = r=l

This completes the proof.

~

S(n - k, r - k) = B . - k .

r=k+l

[]

E x a m p l e 3. We show a 3-tree of 8 vertices in Fig. 3. We now prove a generalization of T h e o r e m 1.

Theorem 3. T h e number o f partitions of {1, ..., n} such that every integer is not in the same part as the previous a integers is Bn-~.

Proof. We use E x a m p l e 3 to visualize the steps of this proof. In Example 3, n = 8 and a = 3. T a k e a set of n vertices, labeled from 1 to n. We form an a-tree from these n vertices, in the following way: for vertex i, 1 ~< i ~< n, draw an edge from vertex i to vertices i - 1, i - 2 , . . . , i - a . Call the resulting graph G. We can think of the hypothesis (that every distinct integer is not in the same part as the previous a distinct integers) in terms of edges and colors; integers i and j are not in the same part (have different colors) iff an edge joins vertices i and j. Therefore we want to consider n o n - i s o m o r p h i c proper colorings of G. By T h e o r e m 2, nipc(G) = B,_,. This completes the proof. [] We note that T h e o r e m 3 is still true if a = 0 or a = n. Equivalently, T h e o r e m 3 is: T h e number of partitions {p(1), . . . , p(n)} of { 1. . . . , n} such that 0 < l i - J l <<.a implies that p(i) ~ p(j) for all 1 ~< i , j <<.n, is B , _ , .

Acknowledgements I a m especially grateful to the S u m m e r U n d e r g r a d u a t e Research Fellowships p r o g r a m (SURF), at Caltech, for its support and funding of this research in the s u m m e r of 1995; and m y adviser Richard M. Wilson, at Caltech, for his encouragement, ideas, and readings of drafts of this paper. I would also like to t h a n k Arthur Benjamin, at H a r v e y M u d d , for c o m m u n i c a t i n g another bijection between restricted partitions of {1 . . . . . n} and unrestricted partitions of {1 . . . . . n - 1}; and D a v i d M.

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Jackson, at the University of Waterloo, for communicating another proof that the number of restricted partitions of {1, ..., n} is B.-1.

References [1] L.W. Beineke and R.E. Pippert, The number of labeted k-dimensional trees, J. Combin. Theory 6 (1969) 200-205. 1-2] P. Erdrs, A. Renyi, and A. T. S6s, (eds.), Combinatorial Theory and its Applications III (Bolyai Jhnos Matematikai Thrsulat, Budapest, Hungary, 1970), 945-946. [3] J.W. Moon, The number of labeled k-trees, J. Combin. Theory 6 (1969) 196--199. 14] R.C. Read, Contributions to the cell growth problem, Canad. J. Math. 14 (1962) 1-20. [51 J.H. van Lint and R.M. Wilson, A Course in Combinatorics (Cambridge University Press, Cambridge, 1992) 20, 105. 1-6] E.G. Whitehead, Enumerative Combinatorics (Courant Institute of Mathematical Sciences, New York University, 1972) 8-13.