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The Deranged Bell Numbers 2 THE DERANGED BELL NUMBERS BELBACHIR HACENE,` DJEMMADA YAHIA, AND NEMETH´ LASZL` O` Abstract. It is known that the ordered Bell numbers count all the ordered partitions of the set [n] = {1, 2,...,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 behavior of the deranged Bell numbers. Finally, we give some brief results for their r-versions. 1. Introduction A permutation σ of a finite set [n] := {1, 2,...,n} is a rearrangement (linear ordering) 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) 6= i for all (1 ≤ i ≤ n) (fixed-point-free permutation). The derangement number dn denotes the number of all derangements of the set [n]. A simple combinatorial approach yields the two recursions for dn (see for instance [13]) dn = (n − 1)(dn−1 + dn−2) (n ≥ 2) and n dn = ndn−1 + (−1) (n ≥ 1), with the first values d0 = 1 and d1 = 0. The derangement number satisfies the explicit expression (see [3]) n (−1)i d = n! . n i! i=0 X The generating function of the sequence dn is given by tn e−t D(t)= dn = . arXiv:2102.00139v1 [math.GM] 30 Jan 2021 n! 1 − t n≥ X0 The first few values of dn are (dn)n≥0 = {1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961,...}. For more details about derangement numbers we refer readers to [6, 13, 19] and the references therein. A partition of a set [n] := {1, 2,...,n} is a distribution of their elements to k non-empty disjoint subsets B1|B2| ... |Bk called blocks. We assume that the blocks are arranged in ascending order according to their minimum elements (min B1 < min B2 < ··· < min Bk). 2000 Mathematics Subject Classification. Primary: 11B73; Secondary: 05A18, 05A05. 1 THE DERANGED BELL NUMBERS 2 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˜ = .
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