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n It is well-known that the Stirling numbers of the second kind, denoted k , count the partitions’ number of the set [n] into k non-empty blocks. The numbers n  k satisfy the recurrence  n n − 1 n − 1 = + k (1 ≤ k ≤ n), k k − 1 k       n n with 0 = δn,0 (Kronecker delta) and k = 0 (k>n). An ordered partition ψ of a set [n] is a permutation σ of the partition B1|B2| ... |Bk, in other words, we consider all the orders of the blocks,

ψ([n]) = Bσ(1)|Bσ(2)| ... |Bσ(k). For notation, throughout this paper we represent the elements of the same block by adjacent numbers and we separate the blocks by bars ”|”. Example 1. The partitions of the set [3] = {1, 2, 3} are: 123;1|23;12|3;13|2; 1|2|3, 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|2|3; 1|3|2; 2|1|3; 2|3|1; 3|1|2; 3|2|1. The total number of the ordered partitions of the set [n] is known as the ordered Bell number or Fubini number [7, 14], denoted by Fn, which is given by n n F = k! . n k k X=0   The first few values of the ordered Bell numbers are

(Fn)n≥0 = {1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563,...}. The explicit formula for the Stirling number of the second kind k n 1 k = (−1)k−j jn k k! j j=0   X   follows the explicit formula for the ordered Bell number n k k F = (−1)k−j jn. n j k j=0 X=0 X   The exponential generating function for Fn is given by tn 1 (1.1) F(t)= F = . n n! 2 − et n≥ X0 We note that if the order of the blocks does not matter, then the total number of partitions of a set [n] is given by Bell numbers n n B = . n k k X=0   The exponential generating function for Bn is n t t (1.2) B(t)= B = ee −1. n n! n≥ X0 Most of the previous works focused on the some restrictions and generalizations of Stirling number of second kind to introduce new classes of ordered Bell number (see [5] and the references given there). THE DERANGED BELL NUMBERS 3

The aim of our paper is to introduce and study a new classes of ordered partitions numbers by taking into account the derangement of blocks (or permutations without fixed blocks).

2. The deranged Bell numbers In this section, we introduce the notion of deranged partition and we study the deranged Bell numbers.

Definition 1. A deranged partition ψ˜ of the set [n] is a derangementσ ˜ of the partition B1|B2| ... |Bk, i.e.,

˜ ψ([n]) = Bσ˜(1)|Bσ˜(2)| ... |Bσ˜(k) such that Bσ˜(i) 6= Bi for all (1 ≤ i ≤ k).

Definition 2. Let F˜n be the deranged Bell number which counts the total number of the deranged partitions of the set [n].

Proposition 2.1. For all n ≥ 0 we have that

n n (2.1) F˜ = d . n k k k X=0   n Proof. Since counts the number of partitions of [n] into k blocks, then the k   n number of deranged partitions of [n] having k blocks is d (derangement of k k n  n blocks). Therefore the nth deranged Bell number is F˜ = d .  n k k k X=0  

Here are the first few values of F˜n:

(F˜n)n≥0 = {1, 0, 1, 5, 28, 199, 1721, 17394, 200803, 2607301, 37614922,...}.

In Tables 1 and 2, we give few examples of the deranged permutations.

Set Partition b1|b2| · · · |bk Deranged partitions F˜n 123 ∅ 1|23 23|1 {1,2,3} 12|3 3|12 5 13|2 2|13 1|2|3 2|3|1 3|1|2 Table 1. Deranged partitions of the set [3]. THE DERANGED BELL NUMBERS 4

Set Partition b1|b2| · · · |bk Deranged partitions F˜n 1234 ∅ 1|234 234|1 12|34 34|12 134|2 2|134 123|4 4|123 14|23 23|14 124|3 3|124 13|24 24|13 {1, 2, 3, 4} 1|2|34 2|34|1 34|1|2 28 1|23|4 23|4|1 4|1|23 1|24|3 24|3|1 3|1|24 12|3|4 3|4|12 4|12|3 13|2|4 2|4|13 4|13|2 14|2|3 2|3|14 3|14|2 2|1|4|3 2|3|4|1 2|4|1|3 1|2|3|4 3|1|4|2 3|4|1|2 3|4|2|1 4|1|2|3 4|3|1|2 4|3|2|1

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 is given by t tn e−(e −1) F˜(t)= F˜ = . n n! 2 − et n≥ X0

Proof. Denote by F˜(t) the exponential generating function of the sequence F˜n. From (2.1) we have n t n tn n tn (et − 1)k e−(e −1) F˜(t)= d = d = d = . k k n! k k n! k k! 2 − et n≥ k k≥ n≥ k≥ X0 X=0   X0 X0   X0 

3.2. Explicit formula.

Theorem 3.2. For any n ≥ 0, the sequence F˜n can be expressed explicitly as

n k (−1)k+i−j k F˜ = jn. n i! j k i,j=0 X=0 X   Proof. From the explicit formulas of Stirling numbers of the second kind and de- rangement number we have THE DERANGED BELL NUMBERS 5

n n k k n k n (−1)i 1 k (−1)k+i−j k F˜ = d = k! (−1)k−j jn = jn. n k k i! k! j i! j k k i=0 j=0 k i,j=0 X=0   X=0 X X   X=0 X   

3.3. Dobi`nski’s formula. One of the most important result for Bell number was established by Dobi`nski [8, 9, 17], where he expressed wn in the infinite series form bellow 1 kn w = . n e k! k≥ X0 An analogue result for the ordered Bell number was given by Gross [12] as 1 kn F = . n 2 2k k≥ X0

The Dobi`nski’s formula for F˜n is established by our next theorem Theorem 3.3. For any n ≥ 0 we have e (−1)j kn F˜ = . n 2 j! 2k−j j≥ k≥j X0 X Proof. From Theorem 3.1 it follows that

t t tn e−(e −1) e e−e F˜ = = t n n! 1 − (et − 1) 2 e n≥ 1 − 2 X0 e (−1)kekt 1 k
= ekt 2 k! 2 k≥ k≥ X0 X0   k e (−1)jejt 1 k−j = e(k−j)t 2 j! 2 k≥ j=0 X0 X   k e (−1)j 1 k−j tn = kn 2 j! 2 n! n≥ k≥ j=0 X0 X0 X   e (−1)j 1 k−j tn = kn , 2 j! 2 n! n≥ j≥ k≥j X0 X0 X   tn by comparing the coefficient of n! we get e (−1)j kn F˜ = . n 2 j! 2k−j j≥ k≥j X0 X 

Remark 3.4. Dobi`niski’s formula is suitable to the computation of wn, Fn 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. THE DERANGED BELL NUMBERS 6

th 1 Lemma 4.1. For any m ≥ 1, the m derivatives of F(t) and W(t) are, respectively,

m m (4.1) F (m)(t)= k! ektF k+1(t) k k X=0   and

m 1 (m) m ekt (4.2) = (−1)k . W(t) k W(t) k   X=0   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

1 1 (4.3) F ′(t)= etF 2(t)= k! ektF k+1(t) k k X=0   and

1 ′ −et 1 1 ekt (4.4) = = (−1)k . W(t) W(t) k W(t) k   X=0   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

m m F (m)(t)= k! ektF k+1(t) k k X=0   and m 1 (m) m ekt = (−1)k . W(t) k W(t) k   X=0   Now, we prove the statement for m + 1. Thus,

m ′ m F (m+1)(t)= k! ektF k+1(t) k k ! X=0   m m = k! kektF k+1(t) + (k + 1)ektF k(t)F ′(t) k k=0   Xm  m  m m = k! kektF k+1(t)+ (k + 1)! e(k+1)tF k+2(t) k k k k X=0   X=0   m m +1 m +1 m = k! kektF k+1(t)+ k! ektF k+1(t) k k − 1 k k X=0   X=0   m +1 m m = k! k + ektF k+1(t) k k − 1 k X=0      m+1 m +1 = k! ektF k+1(t) k k X=0   THE DERANGED BELL NUMBERS 7 and m ′ 1 (m) m ekt = (−1)k W(t) k W(t)   k=0   ! mX m ekt(kW(t) −W′(t)) = (−1)k k W2(t) k=0    Xm m m ekt m e(k+1)t = (−1)k k + (−1)k+1 k W(t) k W(t) k k X=0   X=0   m m +1 m ekt +1 m ekt = (−1)k k + (−1)k k W(t) k − 1 W(t) k k X=0   X=0   m +1 m m ekt = (−1)k k + k k − 1 W(t) k X=0      m +1 m +1 ekt = (−1)k . k W(t) k X=0   Therefore, the assumption holds true for m + 1, which complete the proof.  We can now formulate the higher order derivative for F˜(t). First of all, we define the so-called ith falling factorial of j by i(i − 1)(i − 2) ··· (i − j + 1), if i ≥ 0; ji = ( 1, if i =0. Theorem 4.2. For any m ≥ 1 we have m k m−k m k m − k (4.5) F˜(m)(t)= F˜(t) (−1)j i! e(j+i)tF i(t) k i j k i=0 j=0 X=0   X X    or equivalently m m m (4.6) F˜(m)(t)= F˜(t) (−1)i+j ejtji F i(t), j i=0 j=i X X   where F˜(m)(t) is the mth 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]) n n (f(t)g(t))(n) = f (k)(t)g(n−k)(t) k k X=0   we have m F(t) (m) m 1 (m−k) F˜(m)(t)= = F (k)(t) . W(t) k W(t) k   X=0     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 m k m − k j m = , k i j − i i j k i X=        THE DERANGED BELL NUMBERS 8 a simple calculation gives the result.  Let us give an important consequence of the preceding theorem. Corollary 4.3. For any n ≥ 0, the nth deranged Bell number satisfies the following binomial convolution − n i 1 n i F˜ = F˜ F . n+1 i j j i−j i=0 j=0 X X    Proof. To deduce the result from (4.5) for m = 1 we have F˜′(t)= etF˜(t) (F(t) − 1)

n n n t t t t = e F˜n Fn − F˜n  n! n! n!  n≥ n≥ n≥ X0 X0 X0   n−1 n t n t = e F˜j Fn−j  j  n! n≥ j=0 X0 X   ni−1  n i tn = F˜ F − . i j j i j n! n≥ i=0 j=0 X0 X X    In another hand, it is easy to check that tn F˜′(t)= F˜ . n+1 n! n≥ X0 Therefore, the corollary holds true.  As a more general result we have Corollary 4.4. For any n ≥ 0, the nth deranged Bell number satisfies the following multinomial convolution m m n m i+2 ˜ i+j i k1 ˜ Fn+m = (−1) j j Fk2 Fks . k1, k2,...,ki+2 j i=0 j=i k k ··· ki n s=3 X X 1+ 2+X+ +2=     Y Proof. By applying generalized Cauchy product rule on identity (4.6) and compar- tn  ing the coefficients of n! we get the convolution. Corollary 4.5. For all n ≥ 1 we have n n n n F˜ F = (−1)n−j F˜ . j n−j j j j+1 j=1 j=0 X   X   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. 

5. 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 (an)n≥0 means to find a second sequence bn simple than an which gives a good approximation of its values when n is large. We will use the classical singularity analysis technic (see for instance [11] and Chapter 5 of [21]) to deduce the asymptotic behavior a sequence an using the singularities of its generating function A(t). THE DERANGED BELL NUMBERS 9

Theorem 5.1. The asymptotic behavior F˜n is F˜ 1 n ∼ + O 6.3213)−n , n −→ ∞. n! 2e logn+1(2)
Proof. We can summarize the singularity analysis technic in the following steps: • Compute the singularities of A(t). • Compute the dominant singularity χ0 (singularity of smallest modulus). • Compute the residue of A(t) at χ0

Res(A(t); t = χ0) = lim (t − χ0)A(t). t→χ0 • The generating function A(t) satisfies Res(A(t); t = χ ) A(t) ∼W(t)= 0 . (t − χ0) • By comparing Taylor series coefficients of tn tn A(t)= a and C(t)= c . n n! n n! n≥ n≥ X0 X0 We get the asymptotic behavior an when n is big enough given by −n an ∼ cn + O(ρ ), n −→ ∞, where ρ is the modulus of the next-smallest modulus singularity. t ˜ e−(e −1) Now, applying the previous steps on the generating function F(t) = 2−et , the singularities of F˜(t) are χk = log(2)+2kiπ. The dominant singularity is χ0 = log(2) and the residue at this point is 1 Res(F˜(t),t = χ0) = lim (t − χ0)F˜(t)= − . t→χ0 2e Thus 1 1 tn F˜(t) ∼ = . 2e(log(2) − t) 2e n+1 n≥ log (2) X0 Therefore the asymptotic behavior F˜n is F˜ 1 n ∼ + O(ρ−n), n −→ ∞, n! 2e logn+1(2) where ρ = log2(2) + (2iπ)2 ≃ 6.3213.  q 6. The r-deranged Bell number The r-version of special numbers is a common natural extension in enumerative combinatorics, 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 numbers n of the second kind, denoted k r, count the number of partitions π of the set [n] having exactly k blocks such that the r first elements 1, 2,...,r must  be in distinct blocks. The r-Stirling numbers of the second kind kind have the following gen- erating function [4] n + r tn ert(ez − 1)k = , k + r n! k! n≥ r X0   THE DERANGED BELL NUMBERS 10

and their explicit formula [15] is k n + r 1 k = (−1)k−j (j + r)n. k + r k! j r j=0   X   • The r-derangement numbers introduced by Wang et al [19]. The r-derangement numbers, denoted dn,r, count the number of permutations of the set [n + r] having no fixed points such that the r first elements 1, 2,...,r must be in distinct cycles. The exponential generating function for r-derangement numbers is tn tre−t d = , n,r n! (1 − t)r+1 n≥ X0 and their explicit formula is n i n! d = (−1)n−i,n ≥ r. n,r r (n − i)! i=r X   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-derangementσ ˜ of the set of partitions B1|B2| ... |Br|Br+1| ... |Bk+r, i.e., ˜ Ψ([n + r]) = Bσ˜(1)|Bσ˜(2)| ... |Bσ˜(r)|Bσ˜(r+1)| ... |Bσ˜(k+r) such that Bσ˜(i) 6= Bi for all (1 ≤ i ≤ k + r).

Definition 4. The r-deranged Bell numbers, denoted F˜n,r, count the total number of 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 n n + r F˜ = d . n,r k,r k + r k X=0   6.1. Main properties of the r-deranged Bell numbers. Let us give briefly the main properties of the r-deranged Bell numbers. The proofs are similar to the proves 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 t tn (et(et − 1))r e−(e −1) F˜ = . n,r n! (2 − et)r+1 n≥ X0 • For all positive integers n, k and r, the r-deranged Bell numbers satisfy n k k i k (−1)k+i−j F˜ = (j + r)n, n ≥ r. n,r r j i! k i=r j=0 X=0 X X    • The r-deranged Bell numbers have the following Dobi`nski-like formula r e (−1)k+i−j r j + r F˜ = (2r + k − i)n. n,r 2r+1 2j(k − j)! i j j≥ k≥j i=0 X0 X X    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 THE 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]

L´aszl´oN´emeth University of Sopron, Institute of Mathematics, Sopron, Hungary. and associate member of USTHB, RECITS Laboratory. E-mail: [email protected]