The deranged Bell numbers
aa r X i v : . [ m a t h . G M ] J a n THE DERANGED BELL NUMBERS
BELBACHIR HAC`ENE, DJEMMADA YAHIA, AND N´EMETH L `ASZL `O
Abstract.
It is known that the ordered Bell numbers count all the orderedpartitions of the set [ n ] = { , , . . . , n } . In this paper, we introduce the de-ranged Bell numbers that count the total number of deranged partitions of [ n ].We first study the classical properties of these numbers (generating function,explicit formula, convolutions, etc.), we then present an asymptotic behaviorof the deranged Bell numbers. Finally, we give some brief results for their r -versions. Introduction A permutation σ of a finite set [ n ] := { , , . . . , n } is a rearrangement (linearordering) of the elements of [ n ], and we denote it by σ ([ n ]) = σ (1) σ (2) · · · σ ( n ) . A derangement is a permutation σ of [ n ] that verifies σ ( i ) = i for all (1 ≤ i ≤ n )(fixed-point-free permutation). The derangement number d n denotes the numberof all derangements of the set [ n ]. A simple combinatorial approach yields the tworecursions for d n (see for instance [13]) d n = ( n − d n − + d n − ) ( n ≥ d n = nd n − + ( − n ( n ≥ , with the first values d = 1 and d = 0.The derangement number satisfies the explicit expression (see [3]) d n = n ! n X i =0 ( − i i ! . The generating function of the sequence d n is given by D ( t ) = X n ≥ d n t n n ! = e − t − t . The first few values of d n are( d n ) n ≥ = { , , , , , , , , , , , . . . } . For more details about derangement numbers we refer readers to [6, 13, 19] andthe references therein.A partition of a set [ n ] := { , , . . . , n } is a distribution of their elements to k non-empty disjoint subsets B | B | . . . | B k called blocks . We assume that the blocksare arranged in ascending order according to their minimum elements (min B < min B < · · · < min B k ). Mathematics Subject Classification.
Primary: 11B73; Secondary: 05A18, 05A05.
It is well-known that the
Stirling numbers of the second kind , denoted (cid:8) nk (cid:9) , countthe partitions’ number of the set [ n ] into k non-empty blocks. The numbers (cid:8) nk (cid:9) satisfy the recurrence (cid:26) nk (cid:27) = (cid:26) n − k − (cid:27) + k (cid:26) n − k (cid:27) (1 ≤ k ≤ n ) , with (cid:8) n (cid:9) = δ n, (Kronecker delta) and (cid:8) nk (cid:9) = 0 ( k > n ).An ordered partition ψ of a set [ n ] is a permutation σ of the partition B | B | . . . | B k ,in other words, we consider all the orders of the blocks, ψ ([ n ]) = B σ (1) | B σ (2) | . . . | B σ ( k ) . For notation, throughout this paper we represent the elements of the same blockby adjacent numbers and we separate the blocks by bars ” | ”. Example 1.
The partitions of the set [3] = { , , } are:123; 1 |
23; 12 |
3; 13 |
2; 1 | | , and its ordered partitions are the permutations of all the partitions above:123; 1 |
23; 23 |
1; 12 |
3; 3 |
12; 13 |
2; 2 |
13; 1 | |
3; 1 | |
2; 2 | |
3; 2 | |
1; 3 | |
2; 3 | | . The total number of the ordered partitions of the set [ n ] is known as the orderedBell number or Fubini number [7, 14], denoted by F n , which is given by F n = n X k =0 k ! (cid:26) nk (cid:27) . The first few values of the ordered Bell numbers are( F n ) n ≥ = { , , , , , , , , , , , . . . } . The explicit formula for the Stirling number of the second kind (cid:26) nk (cid:27) = 1 k ! k X j =0 ( − k − j (cid:18) kj (cid:19) j n follows the explicit formula for the ordered Bell number F n = n X k =0 k X j =0 ( − k − j (cid:18) kj (cid:19) j n . The exponential generating function for F n is given by(1.1) F ( t ) = X n ≥ F n t n n ! = 12 − e t . We note that if the order of the blocks does not matter, then the total number ofpartitions of a set [ n ] is given by Bell numbers B n = n X k =0 (cid:26) nk (cid:27) . The exponential generating function for B n is(1.2) B ( t ) = X n ≥ B n t n n ! = e e t − . Most of the previous works focused on the some restrictions and generalizationsof Stirling number of second kind to introduce new classes of ordered Bell number(see [5] and the references given there).
HE DERANGED BELL NUMBERS 3
The aim of our paper is to introduce and study a new classes of ordered partitionsnumbers by taking into account the derangement of blocks (or permutations withoutfixed blocks). 2.
The deranged Bell numbers
In this section, we introduce the notion of deranged partition and we study thederanged Bell numbers.
Definition 1. A deranged partition ˜ ψ of the set [ n ] is a derangement ˜ σ of thepartition B | B | . . . | B k , i.e.,˜ ψ ([ n ]) = B ˜ σ (1) | B ˜ σ (2) | . . . | B ˜ σ ( k ) such that B ˜ σ ( i ) = B i for all (1 ≤ i ≤ k ). Definition 2.
Let ˜ F n be the deranged Bell number which counts the total numberof the deranged partitions of the set [ n ]. Proposition 2.1.
For all n ≥ we have that (2.1) ˜ F n = n X k =0 d k (cid:26) nk (cid:27) . Proof.
Since (cid:26) nk (cid:27) counts the number of partitions of [ n ] into k blocks, then thenumber of deranged partitions of [ n ] having k blocks is d k (cid:26) nk (cid:27) (derangement ofblocks). Therefore the n th deranged Bell number is ˜ F n = n X k =0 d k (cid:26) nk (cid:27) . (cid:3) Here are the first few values of ˜ F n :( ˜ F n ) n ≥ = { , , , , , , , , , , , . . . } . In Tables 1 and 2, we give few examples of the deranged permutations.Set Partition b | b | · · · | b k Deranged partitions ˜ F n ∅ |
23 23 | { } | |
12 513 | | | | | | | | Table 1.
Deranged partitions of the set [3].
HE DERANGED BELL NUMBERS 4
Set Partition b | b | · · · | b k Deranged partitions ˜ F n ∅ |
234 234 | |
34 34 | | | | | |
23 23 | | | |
24 24 | { , , , } | |
34 2 | | | | | | | | | | | | | | | | | | | |
12 4 | | | | | |
13 4 | | | | | |
14 3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Table 2.
Deranged partitions of the set [4].3.
Fundamental properties
Here are the fundamental properties of the deranged Bell numbers.3.1.
Exponential generating function.Theorem 3.1.
The exponential generating function of deranged Bell numbers isgiven by ˜ F ( t ) = X n ≥ ˜ F n t n n ! = e − ( e t − − e t . Proof.
Denote by ˜ F ( t ) the exponential generating function of the sequence ˜ F n .From (2.1) we have˜ F ( t ) = X n ≥ n X k =0 d k (cid:26) nk (cid:27) t n n ! = X k ≥ d k X n ≥ (cid:26) nk (cid:27) t n n ! = X k ≥ d k ( e t − k k ! = e − ( e t − − e t . (cid:3) Explicit formula.Theorem 3.2.
For any n ≥ , the sequence ˜ F n can be expressed explicitly as ˜ F n = n X k =0 k X i,j =0 ( − k + i − j i ! (cid:18) kj (cid:19) j n . Proof.
From the explicit formulas of Stirling numbers of the second kind and de-rangement number we have
HE DERANGED BELL NUMBERS 5 ˜ F n = n X k =0 d k (cid:26) nk (cid:27) = n X k =0 k ! k X i =0 ( − i i ! 1 k ! k X j =0 ( − k − j (cid:18) kj (cid:19) j n = n X k =0 k X i,j =0 ( − k + i − j i ! (cid:18) kj (cid:19) j n . (cid:3) Dobi`nski’s formula.
One of the most important result for Bell number wasestablished by Dobi`nski [8, 9, 17], where he expressed w n in the infinite series formbellow w n = 1 e X k ≥ k n k ! . An analogue result for the ordered Bell number was given by Gross [12] as F n = 12 X k ≥ k n k . The Dobi`nski’s formula for ˜ F n is established by our next theorem Theorem 3.3.
For any n ≥ we have ˜ F n = e X j ≥ X k ≥ j ( − j j ! k n k − j . Proof.
From Theorem 3.1 it follows that X n ≥ ˜ F n t n n ! = e − ( e t − )1 − ( e t −
1) = e e − e t (cid:0) − e t (cid:1) = e X k ≥ ( − k e kt k ! X k ≥ (cid:18) (cid:19) k e kt = e X k ≥ k X j =0 ( − j e jt j ! (cid:18) (cid:19) k − j e ( k − j ) t = e X n ≥ X k ≥ k X j =0 ( − j j ! (cid:18) (cid:19) k − j k n t n n != e X n ≥ X j ≥ X k ≥ j ( − j j ! (cid:18) (cid:19) k − j k n t n n ! , by comparing the coefficient of t n n ! we get˜ F n = e X j ≥ X k ≥ j ( − j j ! k n k − j . (cid:3) Remark . Dobi`niski’s formula is suitable to the computation of w n , F n and ˜ F n for large n values as Rota mentioned in [17].4. Higher order derivatives and convolution formulas
Before giving our next result, we state the following lemma and proposition.
HE DERANGED BELL NUMBERS 6
Lemma 4.1.
For any m ≥ , the m th derivatives of F ( t ) and W ( t ) are, respectively, (4.1) F ( m ) ( t ) = m X k =0 k ! (cid:26) mk (cid:27) e kt F k +1 ( t ) and (4.2) (cid:18) W ( t ) (cid:19) ( m ) = m X k =0 ( − k (cid:26) mk (cid:27) e kt W ( t ) . Proof.
The proof of lemma proceeds by induction on m .For m = 1 it is easy to check from the generating functions of Fubini numbers(1.1) and Bell numbers (1.2) that(4.3) F ′ ( t ) = e t F ( t ) = X k =0 k ! (cid:26) k (cid:27) e kt F k +1 ( t )and(4.4) (cid:18) W ( t ) (cid:19) ′ = − e t W ( t ) = X k =0 ( − k (cid:26) k (cid:27) e kt W ( t ) . Then from (4.3) and (4.4) the lemma is true for m = 1.Now, assume the lemma holds for a fixed m ≥
1, then F ( m ) ( t ) = m X k =0 k ! (cid:26) mk (cid:27) e kt F k +1 ( t )and (cid:18) W ( t ) (cid:19) ( m ) = m X k =0 ( − k (cid:26) mk (cid:27) e kt W ( t ) . Now, we prove the statement for m + 1. Thus, F ( m +1) ( t ) = m X k =0 k ! (cid:26) mk (cid:27) e kt F k +1 ( t ) ! ′ = m X k =0 k ! (cid:26) mk (cid:27) (cid:2) ke kt F k +1 ( t ) + ( k + 1) e kt F k ( t ) F ′ ( t ) (cid:3) = m X k =0 k ! (cid:26) mk (cid:27) ke kt F k +1 ( t ) + m X k =0 ( k + 1)! (cid:26) mk (cid:27) e ( k +1) t F k +2 ( t )= m +1 X k =0 k ! (cid:26) mk (cid:27) ke kt F k +1 ( t ) + m +1 X k =0 k ! (cid:26) mk − (cid:27) e kt F k +1 ( t )= m +1 X k =0 k ! (cid:18) k (cid:26) mk (cid:27) + (cid:26) mk − (cid:27)(cid:19) e kt F k +1 ( t )= m +1 X k =0 k ! (cid:26) m + 1 k (cid:27) e kt F k +1 ( t ) HE DERANGED BELL NUMBERS 7 and (cid:18) W ( t ) (cid:19) ( m ) = m X k =0 ( − k (cid:26) mk (cid:27) e kt W ( t ) ! ′ = m X k =0 ( − k (cid:26) mk (cid:27) (cid:20) e kt ( k W ( t ) − W ′ ( t )) W ( t ) (cid:21) = m X k =0 ( − k (cid:26) mk (cid:27) k e kt W ( t ) + m X k =0 ( − k +1 (cid:26) mk (cid:27) e ( k +1) t W ( t )= m +1 X k =0 ( − k (cid:26) mk (cid:27) k e kt W ( t ) + m +1 X k =0 ( − k (cid:26) mk − (cid:27) e kt W ( t )= m +1 X k =0 ( − k (cid:18) k (cid:26) mk (cid:27) + (cid:26) mk − (cid:27)(cid:19) e kt W ( t )= m +1 X k =0 ( − k (cid:26) m + 1 k (cid:27) e kt W ( t ) . Therefore, the assumption holds true for m + 1, which complete the proof. (cid:3) We can now formulate the higher order derivative for ˜ F ( t ). First of all, we definethe so-called i th falling factorial of j by j i = ( i ( i − i − · · · ( i − j + 1) , if i ≥ , if i = 0 . Theorem 4.2.
For any m ≥ we have (4.5) ˜ F ( m ) ( t ) = ˜ F ( t ) m X k =0 (cid:18) mk (cid:19) k X i =0 m − k X j =0 ( − j i ! (cid:26) ki (cid:27)(cid:26) m − kj (cid:27) e ( j + i ) t F i ( t ) or equivalently (4.6) ˜ F ( m ) ( t ) = ˜ F ( t ) m X i =0 m X j = i ( − i + j e jt j i (cid:26) mj (cid:27) F i ( t ) , where ˜ F ( m ) ( t ) is the m th derivative of ˜ F ( t ) .Proof. From the generating function (Theorem 3.1), it is easy to observe that˜ F ( t ) = F ( t ) W ( t ) . According to Leibniz’s formula (see section 5 .
11, exercise 4 in [2])( f ( t ) g ( t )) ( n ) = n X k =0 (cid:18) nk (cid:19) f ( k ) ( t ) g ( n − k ) ( t )we have ˜ F ( m ) ( t ) = (cid:18) F ( t ) W ( t ) (cid:19) ( m ) = m X k =0 (cid:18) mk (cid:19) F ( k ) ( t ) (cid:18) W ( t ) (cid:19) ( m − k ) . And from the precedent Lemma (equations (4.1) and (4.2)), we get the identity(4.5).For the equivalent identity (4.6), we use the equation [16, page 120] m X k = i (cid:18) mk (cid:19)(cid:26) ki (cid:27)(cid:26) m − kj − i (cid:27) = (cid:18) ji (cid:19)(cid:26) mj (cid:27) , HE DERANGED BELL NUMBERS 8 a simple calculation gives the result. (cid:3)
Let us give an important consequence of the preceding theorem.
Corollary 4.3.
For any n ≥ , the n th deranged Bell number satisfies the followingbinomial convolution ˜ F n +1 = n X i =0 i − X j =0 (cid:18) ni (cid:19)(cid:18) ij (cid:19) ˜ F j F i − j . Proof.
To deduce the result from (4.5) for m = 1 we have˜ F ′ ( t ) = e t ˜ F ( t ) ( F ( t ) − e t X n ≥ ˜ F n t n n ! X n ≥ F n t n n ! − X n ≥ ˜ F n t n n ! = e t X n ≥ n − X j =0 (cid:18) nj (cid:19) ˜ F j F n − j t n n != X n ≥ n X i =0 i − X j =0 (cid:18) ni (cid:19)(cid:18) ij (cid:19) ˜ F j F i − j t n n ! . In another hand, it is easy to check that˜ F ′ ( t ) = X n ≥ ˜ F n +1 t n n ! . Therefore, the corollary holds true. (cid:3)
As a more general result we have
Corollary 4.4.
For any n ≥ , the n th deranged Bell number satisfies the followingmultinomial convolution ˜ F n + m = m X i =0 m X j = i X k + k + ··· + k i +2 = n (cid:18) nk , k , . . . , k i +2 (cid:19) ( − i + j j i (cid:26) mj (cid:27) j k ˜ F k i +2 Y s =3 F k s . Proof.
By applying generalized Cauchy product rule on identity (4.6) and compar-ing the coefficients of t n n ! we get the convolution. (cid:3) Corollary 4.5.
For all n ≥ we have n X j =1 (cid:18) nj (cid:19) ˜ F n − j F j = n X j =0 ( − n − j (cid:18) nj (cid:19) ˜ F j +1 . Proof.
The result holds true by applying the well-known binomial inversion formula(see for example [1, Corollary 3.38, p. 96]) on Corollary 4.3. (cid:3) Asymptotic behavior ˜ F n In this section, we are interested to obtaining the asymptotic behavior the de-ranged Bell numbers ˜ F n .Finding an asymptotic behavior of a sequence ( a n ) n ≥ means to find a secondsequence b n simple than a n which gives a good approximation of its values when n is large.We will use the classical singularity analysis technic (see for instance [11] andChapter 5 of [21]) to deduce the asymptotic behavior a sequence a n using thesingularities of its generating function A ( t ). HE DERANGED BELL NUMBERS 9
Theorem 5.1.
The asymptotic behavior ˜ F n is ˜ F n n ! ∼ e log n +1 (2) + O (cid:0) . − n (cid:1) , n −→ ∞ . Proof.
We can summarize the singularity analysis technic in the following steps: • Compute the singularities of A ( t ). • Compute the dominant singularity χ (singularity of smallest modulus). • Compute the residue of A ( t ) at χ Res ( A ( t ); t = χ ) = lim t → χ ( t − χ ) A ( t ) . • The generating function A ( t ) satisfies A ( t ) ∼ W ( t ) = Res ( A ( t ); t = χ )( t − χ ) . • By comparing Taylor series coefficients of A ( t ) = X n ≥ a n t n n ! and C ( t ) = X n ≥ c n t n n ! . We get the asymptotic behavior a n when n is big enough given by a n ∼ c n + O ( ρ − n ) , n −→ ∞ , where ρ is the modulus of the next-smallest modulus singularity.Now, applying the previous steps on the generating function ˜ F ( t ) = e − ( et − − e t , thesingularities of ˜ F ( t ) are χ k = log (2) + 2 kiπ .The dominant singularity is χ = log (2) and the residue at this point is Res ( ˜ F ( t ) , t = χ ) = lim t → χ ( t − χ ) ˜ F ( t ) = − e . Thus ˜ F ( t ) ∼ e (log(2) − t ) = 12 e X n ≥ t n log n +1 (2) . Therefore the asymptotic behavior ˜ F n is˜ F n n ! ∼ e log n +1 (2) + O ( ρ − n ) , n −→ ∞ , where ρ = q log (2) + (2 iπ ) ≃ . . (cid:3) The r -deranged Bell number The r -version of special numbers is a common natural extension in enumerativecombinatorics, see, for example, the r -Stirling numbers [4], r -Bell numbers [15], r -Fubini numbers [5], r -derangement numbers [19].The motivation of this section came from two earlier researches: • The r -Stirling numbers introduced by Border [4]. The r -Stirling numbersof the second kind, denoted (cid:8) nk (cid:9) r , count the number of partitions π of theset [ n ] having exactly k blocks such that the r first elements 1 , , . . . , r mustbe in distinct blocks.The r -Stirling numbers of the second kind kind have the following gen-erating function [4] X n ≥ (cid:26) n + rk + r (cid:27) r t n n ! = e rt ( e z − k k ! , HE DERANGED BELL NUMBERS 10 and their explicit formula [15] is (cid:26) n + rk + r (cid:27) r = 1 k ! k X j =0 ( − k − j (cid:18) kj (cid:19) ( j + r ) n . • The r -derangement numbers introduced by Wang et al [19]. The r -derangementnumbers, denoted d n,r , count the number of permutations of the set [ n + r ]having no fixed points such that the r first elements 1 , , . . . , r must bein distinct cycles. The exponential generating function for r -derangementnumbers is X n ≥ d n,r t n n ! = t r e − t (1 − t ) r +1 , and their explicit formula is d n,r = n X i = r (cid:18) ir (cid:19) n !( n − i )! ( − n − i , n ≥ r. Now, it’s natural to define the r -deranged Bell numbers as Definition 3. An r -deranged partition ˜Ψ of the set [ n + r ] is an r -derangement ˜ σ of the set of partitions B | B | . . . | B r | B r +1 | . . . | B k + r , i.e.,˜Ψ([ n + r ]) = B ˜ σ (1) | B ˜ σ (2) | . . . | B ˜ σ ( r ) | B ˜ σ ( r +1) | . . . | B ˜ σ ( k + r ) such that B ˜ σ ( i ) = B i for all (1 ≤ i ≤ k + r ). Definition 4.
The r -deranged Bell numbers , denoted ˜ F n,r , count the total numberof the deranged partitions of the set [ n + r ].It’s clear that, for all positive integers, n , k and r with ( r ≤ k ≤ n ) , we have˜ F n,r = n X k =0 d k,r (cid:26) n + rk + r (cid:27) . Main properties of the r -deranged Bell numbers. Let us give brieflythe main properties of the r -deranged Bell numbers. The proofs are similar to theproves of previous results, so we leave the verifications to the readers. • For all positive integers n , k and r , the exponential generating function of˜ F n,r is X n ≥ ˜ F n,r t n n ! = ( e t ( e t − r e − ( e t − )(2 − e t ) r +1 . • For all positive integers n , k and r , the r -deranged Bell numbers satisfy˜ F n,r = n X k =0 k X i = r k X j =0 (cid:18) ir (cid:19)(cid:18) kj (cid:19) ( − k + i − j i ! ( j + r ) n , n ≥ r. • The r -deranged Bell numbers have the following Dobi`nski-like formula˜ F n,r = e r +1 X j ≥ X k ≥ j r X i =0 ( − k + i − j j ( k − j )! (cid:18) ri (cid:19)(cid:18) j + rj (cid:19) (2 r + k − i ) n . Here are the first few r -deranged Bell numbers. n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 r = 0 1 0 1 5 28 199 1721 17394 r = 1 1 5 28 199 1721 17394 200803 2607301 r = 2 2 30 362 4390 56912 801668 12289342 204429498 r = 3 6 180 3810 72960 1377936 26643204 536553870 11341749600 r = 4 24 1200 39360 1099560 28812504 741799296 19236973920 509589280200 r = 5 120 9000 422520 16237200 565687080 18805154760 614116782840 20053080534960 r = 6 720 75600 4808160 243341280 10892100240 455188401360 18332566132320 725927285809440 HE DERANGED BELL NUMBERS 11
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Hac`ene Belbachir
USTHB, Faculty of Mathematics,RECITS Laboratory,BP 32, El Alia, 16111,Bab Ezzouar, Algiers, Algeria.E-mail: [email protected] or [email protected]
Yahia Djemmada
USTHB, Faculty of Mathematics,RECITS Laboratory,BP 32, El Alia, 16111,Bab Ezzouar, Algiers, Algeria.E-mail: [email protected] or [email protected]