Sets of iterated Partitions and the Bell iterated Exponential Integers
aa r X i v : . [ m a t h . C O ] M a r Sets of iterated Partitions and theBell iterated Exponential Integers
Ivar Henning SkauUniversity of South-Eastern Norway3800 Bø, [email protected] Forsberg KristensenUniversity of South-Eastern Norway3918 Porsgrunn, [email protected] 21, 2019
Abstract
It is well known that the Bell numbers represent the total numberof partitions of an n-set. Similarly, the Stirling numbers of the secondkind, represent the number of k-partitions of an n-set. In this paper weintroduce a certain partitioning process that gives rise to a sequence ofsets of ”nested” partitions. We prove that at stage m, the cardinalityof the resulting set will equal the m-th order Bell number. This set-theoretic interpretation enables us to make a natural definition of higherorder Stirling numbers and to study the combinatorics of these entities.The cardinality of the elements of the constructed ”hyper partition” setsare explored.
Consider the 3-set S = { a, b, c } . The partition set ℘ (1)3 of S , where the elementsof S are put into boxes, contains the five partitions shown in the second columnof Figure 1.Now we proceed, putting boxes into boxes. This means that we create a secondorder partition set to each first order partition in ℘ (1)3 . The union of all thesecond order partition sets is denoted by ℘ (2)3 , and appears in the third columnof Figure 1. 1 = ℘ (0)3 ℘ (1)3 ℘ (2)3 { abc ,ab c ,ac b ,bc a ,a b c }{ a, b, c } { abc ,ab c , ab c ,ac b , ac b ,bc a , bc a ,a b c , a b c ,a c b , b c a , a b c } Figure 1: The basic set S together with the partition sets ℘ (1)3 and ℘ (2)3 Definition. ℘ (1) n is the set of all partitions of a given n -set. For m > ℘ ( m ) n ,called the m -th order partition set, is the union of the complete collection ofsets, each being the partition set of an element in ℘ ( m − n .We observe that the number of partitions in ℘ (2)3 is | ℘ (2)3 | = 12, which also, andnot by coincidence, turns out to be the second order Bell number B (2)3 . The m -th order Bell numbers B ( m ) n ( n = 0 , , . . . ), studied by E. T. Bell in [1],are given by the exponential generating functions E m ( x ) = ∞ X n =0 B ( m ) n x n n ! ( m ≥ , where E ( x ) = exp(exp( x ) −
1) and E m +1 ( x ) = exp( E m ( x ) − B ( m ) n is computed for a few values of m and n . Theorem 1.
The number of m -th order partitions of an n -set is B ( m ) n ( m, n ≥ ), i.e. | ℘ ( m ) n | = B ( m ) n . (1)2 \ n B ( m ) n when 1 ≤ n ≤ ≤ m ≤ Proof.
The proof makes use of generating functions. Recall that there are alto-gether S ( n, k ) (Stirling number of the second kind) distinct k -partitions of thegiven n -set, i.e. there are S ( n, k ) elements in ℘ (1) n which are k -sets. Each ofthese k -sets gives rise to | ℘ ( m ) k | distinct partitions of order m + 1 of the n -set westarted with, i.e. elements in ℘ ( m +1) n . Furthermore, different elements in ℘ (1) n ofcourse give different elements in ℘ ( m +1) n , since they are already different at the”ground level”. This means that we have the recurrence formula | ℘ ( m +1) n | = n X k =1 | ℘ ( m ) k | S ( n, k ) , | ℘ (0) k | = 1 . (2)Now, let P ( m ) ( x ) = P ∞ n =1 | ℘ ( m ) n | · x n /n ! denote the exponential generating func-tion of {| ℘ ( m ) n |} ∞ n =1 . Multiplication with x n /n !, summation over n and changingthe order of summation in (2), leads to P ( m +1) ( x ) = ∞ X k =1 | ℘ ( m ) k | ∞ X n =1 S ( n, k ) · x n n ! . (3)It is well known that P ∞ n =1 S ( n, k ) · x n /n ! = ( e x − k /k ! is the exponentialgenerating function of the Stirling numbers of the second kind. (A proof of thisfact is included in Example 4.) From (3) we therefore get the recurrence formula P ( m +1) ( x ) = ∞ X k =1 | ℘ ( m ) k | · ( e x − k k ! = P ( m ) ( e x − . (4)Now, since P (0) ( x ) = e x −
1, a straightforward induction argument yields P ( m ) ( x ) = E m ( x ) −
1, which proves (1) as well as the relation B ( m +1) n = n X k =1 B ( m ) k S ( n, k ) , B (0) k = 1 , (5)which now follows from (2). 3 xample 1. To demonstrate the power of hyper partition thinking, we give acombinatorial proof of the relation B ( m ) n = n − X s =0 (cid:18) n − s (cid:19) B ( m ) s B ( m − n − s , B ( m )0 = 1 , (6)that appeared in [1, p. 545, (2.11)].Each element (i.e. a hyper partition) in ℘ ( m ) n consists of nested sets where thesets at the ground level are subsets of the basic n -set S . For a partition p ∈ ℘ ( m ) n we call elements of S related if they ”reside” in the same outer set (box) in p .Now, fix an arbitrary a ∈ S . We count the number of partitions in ℘ ( m ) n accord-ing to which elements a is related: Let A be a basic ( n − s )-set, including a , ofrelated elements. Now for each of these (cid:0) n − n − s − (cid:1) = (cid:0) n − s (cid:1) sets there are B ( m − n − s inner structures. For the complementary s -set C = S \ A , the partitioning pro-cess will generate B ( m ) s partitions of order m . By combining the possibilities,(6) follows.Observe that if we in the same manner as above fix two (or more) elementsin S , new formulas emerge.By putting B (0) n = 1, we note that (6) is a generalized version of the wellknown formula (see [3, p. 210]) B n = n − X s =0 (cid:18) n − s (cid:19) B s . Having established the relationship between partitions of order m and higherorder Bell numbers, defining higher order Stirling numbers seems like a naturalthing to do. Definition.
The m -th order Stirling number S ( m ) ( n, k ) (of the second kind) isthe number of k -sets in ℘ ( m ) n .In [2] E. T. Bell gave an analytical definition of what he called generalizedStirling numbers ζ ( k,m ) n by means of generating functions. In Theorem 2 weprove that S ( m ) ( n, k ) = ζ ( k,m ) n .We note that the higher order Stirling numbers S ( m ) ( n, k ) are the entries ofthe matrix S m , where S = ( S ( n, k )). This can be seen by induction from therelation (7) in the proof of Theorem 2. Table 2 is computed with the aid of suchmatrices. 4 S ( m ) (5 , S ( m ) (5 , S ( m ) (5 , S ( m ) (5 , S ( m ) (5 , B ( m )5 Theorem 2.
Let S ( m ) ( n, k ) be the m -th order Stirling numbers of the secondkind. Then we have S ( m ) ( n, k ) = ζ ( k,m ) n , where ζ ( k,m ) n are the generalized Stirling numbers of the second kind, defined byE.T. Bell in [2, p. 91] by the generating functions ( E m − ( t ) − k k ! = X n ζ ( k,m ) n · t n n ! . Proof.
Let F ( m ) k denote the generating function of { S ( m ) ( n, k ) } ∞ n = k . Then wehave F ( m ) k ( x ) ≡ ∞ X n = k S ( m ) ( n, k ) x n n ! . The idea is to come up with an analogous formula to (2), in order to obtain ananalogous formula to (4). We claim that S ( m +1) ( n, k ) = n X i = k S ( m ) ( i, k ) S ( n, i ) . (7)This is true because we know that each of the S ( n, i ) first order partitions willgenerate S ( m ) ( i, k ) k -partitions of order m + 1. Multiplication by x n /n ! followedby summation over n in (7) gives F ( m +1) k ( x ) = X n,i S ( m ) ( i, k ) S ( n, i ) x n n ! = X i S ( m ) ( i, k ) ( e x − i i ! = F ( m ) k ( e x − , because P n S ( n, i ) x n /n ! = ( e x − i /i !. We have thus deduced the recurrenceformula F ( m +1) k ( x ) = F ( m ) k ( e x − . Since F (1) k ( x ) = ( e x − k /k !, induction yields F ( m ) k ( x ) = ( E m − ( x ) − k k ! , ( E ( x ) = e x ) , which completes the proof. 5e notice that Theorem 1 is proved once more since we have | ℘ ( m ) n | = P nk =1 S ( m ) ( n, k )and P ( m ) ( x ) = ∞ X n =1 | ℘ ( m ) n | x n n ! = ∞ X k =1 F ( m ) k ( x ) = E m ( x ) − . How the introduction of the higher order Stirling numbers opens up the scopefor hyper partition thinking, is illustrated in the next examples.
Example 2.
If we, in our construction process of ℘ ( m ) n , stop at the r -th inter-mediate stage, i.e. in ℘ ( r ) n , making up status so far by grouping the elementsaccording to their cardinality before advancing further on, we get the followinggeneralized version of (7): S ( m ) ( n, k ) = n X i = k S ( m − r ) ( i, k ) S ( r ) ( n, i ) , since there are S ( r ) ( n, i ), i -sets in ℘ ( r ) n . This also follows from the matrix rep-resentation S m = S m − r S r , as well as from (7) by induction.Summing from k = 1 to n yields B ( m ) n = n X i =1 B ( m − r ) i S ( r ) ( n, i ) , which generalizes (5). Example 3.
The formula S ( m ) ( n, k ) = n − X s = k − (cid:18) n − s (cid:19) B ( m − n − s S ( m ) ( s, k −
1) (8)may be proved combinatorially in just the same manner as (6) in Example 1.Note that (8) yields (6) by summation over k . Example 4.
Counting the k -sets in ℘ ( m ) n by first forming the k ”families” (outersets) of related elements, we get S ( m ) ( n, k ) = 1 k ! X i + ··· + i k = n (cid:18) ni , . . . , i k (cid:19) B ( m − i · · · B ( m − i k , (9)where in this case B (0) k = 1 , k ≥ B ( m )0 = 0, because a family with i relatives yields B ( m − i elements in ℘ ( m − i , i.e. there are exactly B ( m − i possible ”inner” structures for an ” i -family”.Summing over k yields B ( m ) n = n X k =1 k ! X i + ··· + i k = n (cid:18) ni , . . . , i k (cid:19) B ( m − i · · · B ( m − i k . (10)6ote that we have used nothing but the set-theoretic hyper partition interpre-tation/definition of B ( m ) n and S ( m ) ( n, k ) in establishing (9) and (10). Now,therefore, let f m − ( t ) = f m − be the exponential generating function (e.g.f.) to { B ( m − n } ∞ n =0 . And observe then that f km − is the e.g.f. of the sequence ( X i + ··· + i k = n (cid:18) ni , . . . , i k (cid:19) B ( m − i · · · B ( m − i k ) ∞ n =0 . Now we multiply (9) and (10) with x n /n !, sum over n and change the order ofsummation, to obtain F ( m ) k ( x ) = 1 k ! f km − ( x ) and f m ( x ) = ∞ X k =1 f km − ( x ) k ! = exp( f m − ( x )) − . And since f ( x ) = e x −
1, we have f m ( x ) = E m ( x ) −
1. So by this reasoning”from outside in”, we have an alternative proof of Theorem 1 as well as ofTheorem 2.
In [1, p. 545] E.T. Bell proved the interesting formula B ( m ) n = c n − m n − + c n − m n − + · · · + c , (11)where c n − , . . . , c are rational numbers, independent of m . When n is fixed,this implies that lim m →∞ B ( m ) n B ( m − n = 1 , (12)enabling us to say something more about the cardinality of the members of ℘ ( m ) n . Remark.
Via a slightly different proof of (11) than in [1] one might show that c n − = n ! / n − , see [4].When looking at the ”children” of a p ∈ ℘ ( m ) n , i.e. the elements in ℘ ( m +1) n that p gives rise to, we see that all but one of them have lower cardinality than their”father” p . Following the next generations in the partitioning process, it appearsthat the great majority of the descendants are 1-element sets (see Table 2). Itis therefore easy to conjecture that the average value A ( m ) n of the cardinality ofthe sets in ℘ ( m ) n approaches 1 as m → ∞ , i.e. A ( m ) n = 1 B ( m ) n n X k =1 kS ( m ) ( n, k ) −→ m →∞ . (13)7et us see why (13) is true. (12) yields, in conjunction with the observation S ( m ) ( n,
1) = B ( m − n , that S ( m ) ( n, ∼ B ( m ) n ( m → ∞ ). Since B ( m ) n = P nk =1 S ( m ) ( n, k ), we consequently have S ( m ) ( n, k ) = o ( B ( m ) n ) , k ≥ m → ∞ ) , and (13) follows immediately. The set-theoretic intepretation of the higher order Bell numbers places themin a natural and fundamental context. Together with the higher order Stirlingnumbers, they take the same central position in the combinatorics of the higherorder partition sets as their predecessors B n and S ( n, k ) have at the groundlevel. References [1] Bell, E.T., The Iterated Exponential Integers,
Annals of mathematics,
Vol. 39 , July1938, 539 - 557.[2] Bell, E.T., Generalized Stirling transforms of sequences,
Amer. Journal of Math,
Vol.61 , 1939, 89 - 101.[3] Comtet, L., Advanced Combinatorics,
Reidel , 1974.[4] Skau, I.H., Kristensen, K.F., An asymptotic Formula for the iterated exponential BellNumbers, http://arxiv.org/abs/1903.07979 , 2019., 2019.