Another Family of Permutations Counted by the Bell Numbers
AAnother Family of Permutations Counted by the BellNumbers
Fufa Beyene , Roberto Mantaci b a Addis Ababa University, Addis Ababa, Ethiopia and CoRS; email: [email protected] b IRIF, Universit´e de Paris, Paris, France and CoRS; email: [email protected]
Abstract
Using a permutation code introduced by Rakotondrajao and the second author, we associatewith every set partition of [ n ] a permutation over [ n ], thus defining a class of permutationwhose size is the n -th Bell number. We characterize the permutations belonging to this classand we study the distribution of weak exceedances over these permutations, which turns outto be enumerated by the Stirling numbers of the second kind. We provide a direct bijectionbetween our class of permutations and another equisized class of permutations introduced byPoneti and Vajnovszki. Keywords:
Permutations, Set Partitions, Codes, Subexceedant functions, Exceedances, Bellnumbers, Stirling numbers of the second kind.
1. Introduction
Permutations and set partitions are among the richest objects in combinatorics, they have beenenumerated according to several criteria of interest.There is a plethora of studies of statistics on permutations, many of which are counted byeulerain or macmahonian numbers, such as descents, exceedances, right-to-left minima or max-ima, inversions, etc. On the other hand, we recall the two most basic enumerations for setpartitions : the total number of set partitions of [ n ] is the Bell number [Ro] and the numberof set partitions of [ n ] with k blocks is the Stirling number of the second kind, as given in[Bo, St1]. Corresponding author. a r X i v : . [ m a t h . C O ] J a n oth permutations and set partition can be coded by subexceedant functions, that is, functions f over [ n ] such that 1 ≤ f ( i ) ≤ i for all i ∈ [ n ] (in some contexts it is rather required that0 ≤ f ( i ) ≤ i − π ([Ma]). In this form any integer i ∈ [ n ] is coded with the number of theblock of π where i belongs when π is written in standard form, that is, the elements in eachblock are arranged increasingly and the blocks are arranged in increasing order of their firstelements. In fact, canonical forms of set partitions are restricted growth functions (RGF), aparticular case of subexceedant functions.Several properties of set partitions or permutations can easily be read on their correspondingcodes, this allows to prove elegantly some results by reasoning on the codes rather than on thecoded objects themselves, see for instance the nice article of Baril and Vajnovszki [Ba-Va], andalso the article of Foata and Zeilberger [Fo-Ze].Furthermore, these codes are also useful to implement efficient algorithms for the exhaustivegeneration of the corresponding class of objects. For instance, M. Orlov [Or] used the repre-sentation of set partitions as RGFs to implement an algorithm to generate all set partitions inconstant space and amortized time complexity.F. Rakotondrajao and the second author in [Ma-Ra] defined a new way of coding permutationswith subexceedant functions. This code associates with a subexceedant function f with thepermutation σ = ˜ φ ( f ) defined by the product of transpositions (the leftmost always acts first): σ = ( n f ( n )) ( n − f ( n − · · · (1 f (1)) . We studied further this code in [Be-Ma], where we gave for it a new interpretation based onthe action and the cycle structure of the permutation. In that work we also introduced thedefinition of inom , which will be reminded in the Section 2 and by which we propose to callthis code “ inom code ”.In this paper we associate bijectively a permutation on [ n ] with every set partition of [ n ] bycomposing the canonical form and the inom code. We obtain this way a family of permutationscounted by the Bell numbers and we present some properties and some recurrence relationssatisfied by these objects. 2e show in particular in Section 3 that the permutations belonging to this class can be charac-terised combinatorially and that the distribution of the weak exceedances statistic in this classis the same as the distribution of number of blocks in the set of set partitions, and therefore isgiven by the Stirling numbers of the second kind.In Section 4, we provide a direct bijection between each partition and its corresponding per-mutation (without passing through canonical form and inom code).Finally, in Section 5, we provide a bijection between our family of permutations and anotherBell-counted class of permutations introduced by Poneti and Vajnovszki [Po-Va].
2. Definitions, Notations and Preliminaries
Subexceedant functions
Definition 2.1.
A function f over [ n ] is said to be subexceedant if ≤ f ( i ) ≤ i for all i , ≤ i ≤ n (in some contexts it is required that ≤ f ( i ) ≤ i − ). Notation 2.1.
We denote by F n the set of all subexceedant functions over [ n ] , and if f is afunction over [ n ] , we often denote by f i the value f ( i ) and write f as the word f f . . . f n . Subexceedant functions can be used to code permutations, the Inversion table or the Denerttable are two examples. F. Rakotondrajao and the second author defined in ([Ma-Ra]) a newbijection φ associating to each subexceedant function f a permutation σ = φ ( f ) defined as theproduct of transpositions (the product is from left to right): σ = (1 f ) (2 f ) · · · ( n f n ) . If f i = i , then ( i f i ) = ( i ) does not represent a transposition but the identity permutation.Further, a variation of this bijection associates with a subexceedant function f the permutation σ = ˜ φ ( f ) defined by the product of transpositions (the product is from left to right): σ = ( n f n ) ( n − f n − ) · · · (1 f ) . For example, take f = 121132342. Then φ ( f ) = 568179342 while ˜ φ ( f ) = 497812536.In this paper we will work with the variation ˜ φ .In a previous work ([Be-Ma]) we gave the following :3 efinition 2.2. Let σ = σ (1) σ (2) · · · σ ( n ) ∈ S n . Then the inverse nearest orbital minorant ( inom ) of i ∈ [ n ] is the integer j = σ − t ( i ) ≤ i with t ≥ chosen as small as possible. Example 2.1.
Let σ = 10 6 8 5 1 4 9 3 2 7 = (1 10 7 9 2 6 4 5)(3 8) . Then x inom ( x ) 1 1 3 2 4 2 1 3 7 1This can be clearer with a picture where the permutation is represented as a union of cyclicgraphs. The blue continuous arcs ( i, σ ( i )) represent the action of the permutation, the reddashed arcs ( i, inom ( i )) represent the action of the corresponding subexceedant function. Figure 1: σ and inom Definition 2.3.
We call inom code the bijection ˜ φ − . In ([Be-Ma]), we also proved the following:
Theorem 2.1. If ˜ φ − ( σ ) = f = f f ...f n , then f ( i ) = inom ( i ) . Notation 2.2.
We denote by F n the set of all subexceedant functions over [ n ] , and f = f f ...f n ∈ F n , where f i = f ( i ) , i ∈ [ n ] .Let f = f f ...f n ∈ F n . Then the set of images of f is denoted by Im ( f ) and its cardinality by IM A ( f ) . For instance, in f = 121132342 , then Im ( f ) = { , , , } , IM A ( f ) = 4 . .2. Set Partitions
Definition 2.4.
Let S = [ n ] , the set of the first n positive integers. A set partition π of S is defined as a collection B , . . . , B k of nonempty disjoint subsets such that ∪ ki =1 B i = S . Thesubsets B i will be referred to as ”blocks”. The block representation of a set partition π is: π = B /B / . . . /B k . Definition 2.5.
The block representation of a set partition is said to be standard if the blocks B , . . . , B k are sorted in such way that min ( B ) < min ( B ) < · · · < min ( B k ) and if theelements of every block are arranged in an increasing order. We consider set partitions only in their standard representation. The condition on the orderof the blocks and the arrangement of the integers in each block implies that the standardrepresentation of a set partition is unique and that two different set partitions have differentstandard representations.
Remark 2.1.
Note that in the standard representation, the integer is always in the first block, is in one of the first two blocks, and is in any one of the first three blocks. Indeed, it is easyto show that in the standard representation of a set partition π of [ n ] ,every element i of [ n ] is necessarily in one of the first i blocks. (1) Definition 2.6.
The canonical form of a set partition of [ n ] is a n-tuple indicating the block ofthe standard representation in which each integer occurs, that is, f = (cid:104) f , f , . . . , f n (cid:105) such that j ∈ B f j for all j with ≤ j ≤ n . Example 2.2.
The sequences of canonical forms corresponding to the set partitions of [3] are : (cid:104) , , (cid:105) , (cid:104) , , (cid:105) , (cid:104) , , (cid:105) , (cid:104) , , (cid:105) , (cid:104) , , (cid:105) . Note that canonical form of a set partition is a subexceedant function. But, not all subxeceedantfunctions are canonical forms of set partitions.The set partitions of [ n ] having exactly k blocks are counted by the Stirling numbers of thesecond kind ([Bo, St1]) denoted S ( n, k ), which satisfy the recurrence relation: S ( n, k ) = kS ( n − , k ) + S ( n − , k − n ] is counted by the Bell numbers,denoted B ( n ) and satisfying: B ( n ) = n (cid:88) k =0 S ( n, k ) and B ( n + 1) = n (cid:88) i =0 (cid:18) ni (cid:19) B ( i ) , n ≥ , with B (0) = 1 .
5e denote the set of all set partitions of [ n ] by P ( n ), and its cardinality by B n , (equal to the n -th Bell number B ( n )) with B = 1 (as there is only one set partition of the empty set).In Subsection 3.1 we will associate each set partition of [ n ] with a permutation of the symmetricgroup S n having certain properties. Statistics on permutations
Let σ = σ (1) σ (2) ...σ ( n ) ∈ S n , where S n is the symmetric group of permutations over [ n ].Recall that a weak exceedance of σ is a position i such that σ ( i ) ≥ i ; the set of weak exceedancesof σ is w- Exc ( σ ) = { i : σ ( i ) ≥ i } . The values of weak exceedances are said to be weak exceedanceletters and the subword of σ = σ (1) σ (2) ...σ ( n ) including all weak exceedance letters is denotedby w- ExcL ( σ ). An anti-exceedance of σ is a position i such that σ ( i ) ≤ i , and its value is calledan anti-exceedance letter. Example 2.3.
Let σ = 435129678 . Then w- Exc ( σ ) = { , , , } . The set of weak exceedanceletters of σ , w- ExcL ( σ ) = { σ ( i ) : i ∈ w- Exc ( σ ) } = (cid:104) , , , (cid:105) . In [Ma-Ra] it is proved that if σ = φ ( f ) then the elements in IM A ( f ) correspond to the anti-exceedance letters of σ , while in [Be-Ma], it is proved that if σ = ˜ φ ( f ) then the elements in IM A ( f ) correspond to the weak exceedances (positions) of σ . Remark 2.2.
It is also proved that σ has an anti-exceedance at i if and only if at position i there is the rightmost occurrence of the integer k = f ( i ) in f = φ − ( σ ) . Analoguously σ has aweak exceedance at i if and only if at position k there is the rightmost occurrence of an integer i = f ( k ) in f = ˜ φ − ( σ ) .
3. Inom code and Canonical Forms
In this section we will compare the inom code for permutations with the canonical form of setpartitions of [ n ] and we define a family of permutations counted by the Bell numbers.M. Orlov in [Or] used the following characterization of canonical forms of set partitions toimplement an efficient algorithm for the exhaustive generation of all set partitions for a given n , with constant space and amortised constant time complexity.6 emma 3.1. ([Or]) There is a bijection between all set partitions of the set [ n ] and the set {(cid:104) , k , . . . , k n (cid:105) : k i ∈ N and for all i with ≤ i ≤ n one has ≤ k i ≤ max ≤ j
Let f = f f ...f n be a subexceedant function, then f satisfies the conditionexpressed in equation (2) of Lemma 3.1 if and only if for all i ∈ [ n ] one has { f , f , ..., f i } = { , , ..., p } = [ p ] for a certain p . In other terms, for all i ∈ [ n ] , the set { f , f , ..., f i } is aninteger interval with minimum value 1.Proof. Suppose that f satisfies the condition expressed in equation (2) of Lemma 3.1. We useinduction on i .For i = 1, f = 1 and { } is an interval.Suppose { f , f , ..., f i − } is an interval [ p ]. Then the condition expressed in 2 implies f i ≤ M ax { f , · · · , f i − } ≤ p + 1.So { f , f , ..., f i } = [ p ] ∪ { f i } is either the interval [ p ] or the interval [ p + 1].Conversely, suppose that for all i ∈ [ n ] the set { f , f , ..., f i } is an integer interval [ k ] for some k , let us prove that for all i , f ≤ f i ≤ M ax ≤ j
2. Let { f , f , ..., f i } = [ k ] and { f , f , ..., f i − } = [ k ] for some integers k and k .Then there are only two possibilities: either k = k or k = k + 1.1. Case k = k : this implies that f i ≤ M ax ≤ j
In this subsection, we will present the class permutations associated to
RGF s under ˜ φ . Definition 3.1.
We define Bell permutation of the second kind a permutation σ ∈ S n whosecorresponding subexceedant function f = f f ...f n = ( ˜ φ ) − ( σ ) satisfies : the set { f , f , ..., f i } is an integer interval for all i ∈ [ n ] . We denote by BP ( n ) the set of Bell permutations of the second kind over [ n ] and by b ( n ) itscardinality, the n -th Bell number. Example 3.1.
There are five Bell permutations of the second kind on [3] . These are all thepermutations of S except the permutation . Note that f = ( ˜ φ ) − (213) = 113 and Im ( f ) = { f , f , f } = { , } is not an integer interval. Remark 3.1. If σ = σ (1) σ (2) ...σ ( n ) ∈ S n is a Bell permutation of the second kind, then itsinom code f has IM A ( f ) = { , , . . . , p } for a certain p , therefore σ has weak exceedancesexactly at { , , . . . , p } . Definition 3.2.
Let σ = σ (1) σ (2) ...σ ( n ) ∈ S n . Then the increasing integer sequence Seq ( σ ) associated to σ is given as follows :For x = 1 , , ..., n , we add x to Seq ( σ ) if inom ( x ) = y is not the inom of some integer smallerthan x . Example 3.2.
Let σ = 435129678 = (1 4)(2 3 5)(6 9 8 7) . Then Seq ( σ ) = (cid:104) , , , (cid:105) . Remark 3.2.
The cardinality of
Seq ( σ ) , σ ∈ S n is equal to the cardinality of w- Exc ( σ ) . Thatis, if f = ˜ φ − ( σ ) , then i ∈ Seq ( σ ) if and only if at the position i there is the leftmost occurrenceof the integer k = f ( i ) in f . The characterisation of
RGF s as expressed in Proposition 3.1 implies a characterization forBell permutations of the second kind. 8 heorem 3.1.
Let σ = σ (1) σ (2) ...σ ( n ) ∈ S n with the set of weak exceedance letters w- ExcL ( σ ) = (cid:104) α , α , . . . , α k (cid:105) and Seq ( σ ) = (cid:104) γ , γ , . . . , γ k (cid:105) . Then σ is a Bell-permutation ofthe second kind if and only if the set of the weak exceedances of σ is exactly the interval { , , ..., k } , and γ i ≤ α i , for all i = 1 , , . . . , k. (3) Proof.
Let f ∈ F n be the code of the permutation σ via the bijection ( ˜ φ ) − . As a special caseof Proposition 3.1 for i = n , we have Im ( f ) = { f , f , ..., f n } is an integer interval [ p ] for some p . But on the other hand Im ( f ) coincides with the set of the weak exceedances of σ whosecardinality is k (Remark 2.2). Therefore p = k and the set of the weak exceedances of σ isexactly { , , . . . , k } .Further, for γ i ∈ Seq ( σ ) at position γ i in f we have the value f γ i is the leftmost occurrence in f . Because Seq ( σ ) only includes the positions of the smallest values of all among the values of inoms ( f j = inom ( j )) and for α i ∈ w- ExcL ( σ ) we have f α i is the rightmost occurrence in f (Remark 2.2). From this we see that f = f f ...f n is an integer interval for each j ∈ [ n ] if andonly if w- Exc ( σ ) = [ k ] and the leftmost occurrences of f are increasing. Hence γ i ≤ α i for all i ∈ [ k ]. Example 3.3. Let σ = 763592148 = (1 7)(2 6)(3)(4 5 9 8) . Then Seq ( σ ) = (cid:104) , , , , (cid:105) and w- ExcL ( σ ) = (cid:104) , , , , (cid:105) . Thus the two sequences satisfy (3) and hence σ is a Bellpermutation of the second kind. Let σ = 245987316 = (1 2 4 9 6 7 3 5 8) . Then Seq ( σ ) = (cid:104) , , , , , (cid:105) and w- ExcL ( σ ) = (cid:104) , , , , , (cid:105) . Observe that γ = 8 > α and hence σ is not a Bellpermutation of the second kind. The following proposition gives a recursive procedure to check if a permutation is a Bell Per-mutation of the second kind, for its proof it is convenient to give first the following.
Lemma 3.2.
Let ˜ φ ( f ) = σ ∈ S n , where f is a subexceedant function over [ n ] obtained from thesubexceedant function f (cid:48) over [ n − by concatenating some j ∈ [ n ] at its end, where ˜ φ ( f (cid:48) ) = σ (cid:48) .Then σ is obtained from σ (cid:48) by replacing the integer n by the integer σ (cid:48) ( j ) in σ (cid:48) and appendingthe integer σ (cid:48) ( j ) at the end.If j = n , then σ is obtained from σ (cid:48) by appending n at the end of σ (cid:48) . Proposition 3.2.
A permutation σ = σ (1) σ (2) ...σ ( n ) ∈ S n whose set of weak exceedances isan integer interval [ k ] is in BP ( n ) if and only if the permutation σ (cid:48) ∈ S n − obtained from σ by exchanging the integer n by σ ( n ) in the word σ (1) σ (2) ...σ ( n − is in BP ( n − . roof. According to Lemma 3.2, for all permutations σ , if f = f · · · f n = ( ˜ φ ) − ( σ ) and σ (cid:48) ∈ S n − is the permutation obtained from σ by replacing the integer n by σ ( n ), then thesubexceedant function associated with σ (cid:48) is f (cid:48) = f f · · · f n − .In the hypothesis that w - Exc ( σ ) = { f , ..., f n } is an integer interval [ k ], the two followingconditions become trivially equivalent :1. for all i ∈ [ n ], the set { f , f , ..., f i } is an integer interval with minimum value 1.2. for all i ∈ [ n − { f , f , ..., f i } is an integer interval with minimum value 1.That is, according to Proposition 3.1, σ is Bell if and only if σ (cid:48) is Bell. Example 3.4.
We will look at the three different cases : Consider σ = 7156432 . We have w - Exc ( σ ) = { , , } , which is not an integer interval,thus σ is not a Bell permutation of the second kind. As we have already observed, if one considers σ = 2431 , this permutation has set ofweak exceedances { , , } = [3] but it is not a Bell permutation of the second kind as → and is not Bell permutation of the second kind because its set of weakexceedances, { , } is not an interval. Let σ = 7245613 . We have w - Exc ( σ ) = [5] , so σ may be a Bell permutation of the secondkind. We apply Proposition 3.2 : → → → → . Since is a Bell permutation of the second kind of S we can conclude that σ and thosepermutations obtained in the process are Bell permutations of the second kind.3.2. The distribution of weak exceedances on Bell Permutations of the secondkind
We denote by BP ( n, k ) the set of Bell permutations of the second kind over [ n ] having k weakexceedances and by b ( n, k ) their cardinalities. We can use Proposition 3.2 to give a direct,constructive proof that the cardinalities of the sets BP ( n, k ) equal the Stirling numbers of thesecond kind S ( n, k ).Accordingly, we define the following operation to construct a Bell permutation of the secondkind σ in BP ( n ) starting from a Bell-permutation of the second kind σ (cid:48) ∈ BP ( n −
1) (and aninteger):Let σ (cid:48) ∈ BP ( n − , k ) and i ∈ [ k + 1],then σ is obtained by replacing σ (cid:48) ( i ) by n and then appending σ (cid:48) ( i ) at the end. (4)10 roposition 3.3. The number of Bell permutations of the second kind over [ n ] having k weakexceedances equals the Stirling number of the second kind S ( n, k ) .Proof. We prove that the numbers b ( n, k ) satisfy the same recurrence relation as the Stirlingnumber of the second kind : b ( n, k ) = kb ( n − , k ) + b ( n − , k − . Observe that the operation defined in (4) either preserves the number of weak exceedances orincreases it by 1. Therefore, any Bell permutation of the second kind in BP ( n, k ) can uniquelybe obtained from a permutation σ (cid:48) ∈ BP ( n − , k ) or from a permutation σ (cid:48) ∈ BP ( n − , k − w - Exc ( σ (cid:48) ) = [ k ], i ∈ [ k ] and σ is obtained from σ (cid:48) by the operation defined in (4), then σ ∈ BP ( n, k ). There are b ( n − , k ) possible choices for σ (cid:48) and k possible choices for i hence this contributes kb ( n − , k ) to b ( n, k ).2. if w - Exc ( σ (cid:48) ) = [ k − i = k and σ is obtained from σ (cid:48) by the operation defined in (4),then σ ∈ BP ( n, k ). This contributes b ( n − , k −
1) to the number b ( n, k ).This completes the proof. Example 3.5.
Note that b (4 ,
2) = 2 b (3 ,
2) + b (3 ,
1) = 2 × . So, from each ofthe elements of BP (3 ,
2) = { , , } we obtain 2 Bell permutations of the second kindin BP (4 , by the first operation described in the proof of Proposition 3.3.That is, take [2] × { , , } and apply the first operation to get { , , , , , } ⊆ BP (4 , .Again from BP (3 ,
1) = { } we obtain 1 Bell permutation of the second kind in BP (4 , bythe second operation. That is, take and k = 1 + 1 = 2 . Then by the second operation of theproof of Proposition 3.3 we have { } ⊆ BP (4 , .
4. A bijection between Bell permutations of the second kind and set partitions
Here we give a direct bijection between Bell permutations of the second kind in BP ( n ) andset partitions in P ( n ).Let π = B /B / · · · /B k be a set partition of [ n ] having k blocks. Then by (1) each element of B i is greater than or equal to i , for all i ∈ [ k ].11et σ = σ (1) σ (2) · · · σ ( n ) be a Bell permutation of the second kind with k weak exceedances,then the set of these weak exceedances is w - Exc ( σ ) = [ k ].Define a map λ : BP ( n ) → P ( n ) by λ ( σ ) = π provided that :1. for all weak exceedances ( i, σ ( i )) with i ≥ σ ( i ), insert σ ( i ) in the i -th block, and2. for all non weak exceedance letters i , taken in decreasing order, insert i at the beginningof the inom ( i )-th block. Example 4.1.
Let σ = 45213 ∈ B (5 , . Then λ ( σ ) = π has two blocks. That is, we have / .Observe that inom (3) = 2 , inom (2) = 2 and inom (1) = 1 . So and are in the second block,and is in the first block. Thus we have π = 14 / ∈ P (5 , . Remark 4.1. If λ ( σ ) = π = B / . . . /B k , where w - Exc ( σ ) = [ k ] , then max ( B i ) = σ ( i ) , i =1 , . . . , k . Proposition 4.1.
The map λ is a bijection from BP ( n ) onto P ( n ) .Proof. We will prove that λ is indeed the composition of two bijections: the restriction ofthe permutation code ( ˜ φ ) − to BP ( n ) and the bijection associating to a canonical form itscorrespondent partition.For any σ ∈ BP ( n ), let π ∈ P ( n ) with π = λ ( σ ), let f σ = ( ˜ φ ) − ( σ ) and f π is the canonicalform of π .It suffices to prove that for all Bell-permutations of the second kind σ one has f σ = f π = f λ ( σ ) .Now the definition of ˜ φ implies easily that every integer i is placed in f σ ( i )-th block of π andhence f π ( i ) = f σ ( i ) for all i . Example 4.2.
Take σ = 36821457 ∈ BP (8) . Then λ ( σ ) = π = 13 / / .By applying ˜ φ − to σ we get ˜ φ − ( σ ) = f σ = 12123233 = f π . We now give the direct inverse of the bijection λ .Let us define a mapping χ : P ( n ) (cid:55)→ BP ( n ) such that χ ( π ) = σ , where σ is obtained from π = B /B / . . . /B k ∈ P ( n ) as follows. Let m i = max ( B i ) and σ = σ (1) σ (2) . . . σ ( n ). Then1. For i = 1 , , . . . , k set σ ( i ) = m i .2. For j ∈ [ n ] \{ m , . . . , m k } , in decreasing order, set j = σ ( σ t ( r )), where r is the numberof the block where j is and t is the smallest positive integer such that σ t ( r ) has not yetbeen received an image. Lemma 4.1.
For all π ∈ P ( n ) the image σ = χ ( π ) ∈ BP ( n ) . roof. Let us start by proving w - Exc ( σ ) = [ k ], where k is the number of blocks of π .Note that every element of block B i of π is greater than or equal to i . Hence σ ( i ) = m i ≥ i ,for i = 1 , . . . , k and i ∈ [ k ] is a weak exceedance of σ .Take an integer p > k . We want prove that q = σ ( p ) < p . Let q ∈ B r for some r ≤ k . Then p = σ t ( r ) for a certain t .If t = 1, then p = m r and hence p > q since q is in the r -th block.If t >
1, then the position σ t − ( r ) had arleady received σ ( σ t − ( r )) = σ t ( r ) = p as image. Sincethe integers are inserted in decreasing order we have p > q .Therefore, w - Exc ( σ ) = [ k ] and w - ExcL ( σ ) = (cid:104) α , . . . , α k (cid:105) , where α i = m i and m i is the largestinteger having inom equal to i .By the definition of Seq ( σ ) = (cid:104) γ , . . . , γ k (cid:105) , the integer γ i is the smallest integer having inom equal to i . Therefore, γ i ≤ α i for all i .Hence σ ∈ BP ( n ). Proposition 4.2.
The map χ is the inverse of the map λ .Proof. Let χ ( π ) = σ and λ ( σ ) = π (cid:48) , where π ∈ P ( n, k ). Then we show that π = π (cid:48) .Since w - Exc ( σ ) = [ k ] we have that π (cid:48) ∈ P ( n, k ). That is, π (cid:48) has k blocks.Let x be in the i -th block of π .If x = m i , then σ ( i ) = x and hence x is in the i -th block of π (cid:48) under λ . This is because i ∈ w - Exc ( σ ).If x (cid:54) = m i , then there is the smallest integer t such that x = σ ( σ t ( i )). This means that σ − t − ( x ) = i and hence inom ( x ) = i and x is in the i -th block of π (cid:48) under λ . Thus π = π (cid:48) . Example 4.3.
Let π = 1 4 7 / / / ∈ P (10) . Then σ = σ (1) σ (2) . . . σ ( n ) , where : σ (1) = 7 , σ (2) = 9 , σ (3) = 10 , σ (4) = 8 , and σ (4) = σ (8) , σ (3) = σ (10) , σ (1) = σ (7) , σ (3) = σ (5) , σ (2) = σ (9) , σ (1) = σ (6) . Thus, σ = χ ( π ) = 7 9 10 8 3 1 4 6 2 5 ∈ BP (10) . Our bijection λ allows us to characterise and enumerate some subclasses of BP ( n ) that are asso-ciated to some remarkable subclasses of P ( n ). We give two exemples in the following corollaries. Corollary 4.1.
The number of Bell permutations of the second kind in BP ( n, k ) with σ ( k ) = n and σ ( n ) < k, k (cid:54) = 1 , n equals the total number of Bell permutations of the second kind in BP ( n − , k − . That is, b ( n − , k −
1) = { σ ∈ BP ( n, k ) : σ ( k ) = n, k (cid:54) = 1 , σ ( n ) < k < n } (5)13 roof. We prove that the integer n forms a singleton block in the set P ( n, k ) if and only if thecorresponding Bell permutation of the second kind in BP ( n, k ) satisfies condition 5.Let π ∈ P ( n, k ) and n forms a singleton block in π . Then deleting this block we get π (cid:48) ∈ P ( n − , k − λ − ( π ) = σ and λ − ( π (cid:48) ) = σ (cid:48) . Then σ ∈ BP ( n, k ) and σ (cid:48) ∈ BP ( n − , k −
1) and σ (cid:48) ( k ) < k since σ (cid:48) has only k − σ ∈ BP ( n, k )satisfies condition 5 if and only if σ = ( k n ) · σ (cid:48) . That is, σ ( k ) = n and σ ( n ) = σ (cid:48) ( k ) < k .Note that the number of set partitions in P ( n, k ) in which n forms a singleton block equals b ( n − , k − Remark 4.2.
The equation in 5 is also true for k = 1 or k = n . That is, there is only one setpartition with n blocks which corresponds to the identity permutation, and only one set partitionwith one block which corresponds to n . . . n − . So for all possible k satisfying the conditionof the previous corollary we have all Bell permutations of the second kind over [ n − . Example 4.4.
There are five Bell permutations of the second kind over [4] that satisfy thecondition of Corollary 4.1. These are: , , , , . Their corresponding setpartitions under λ , respectively, are: / , / / , / / , / / , / / / . Removing thesingleton block of from each of them we get all set partitions over [3] .Further, by the corollary we have ↔ , ↔ , ↔ , ↔ , and ↔ . Corollary 4.2.
The number of Bell permutations of the second kind of size n having n − weak exceedances is equal to the number b ( n, n − of set partitions over [ n ] having n − blocks,which, as commonly known, is equal to the sum of the first n − positive integers (cid:0) n (cid:1) = n ( n − .Proof. The second part can be easily demonstrated. Let π ∈ P ( n ) with n − i ∈ [ n −
1] such that the i -th block contains exactly two elements. Since everyelement j ∈ [ n ] must be in any one of the first j blocks, then for all j ∈ [ i ] the integer j is inthe j -th block, hence the second element of block i can only be chosen from [ i + 1 , n ], thereforewe have n − i possible ways to select the other element in to the i -th block. This implies thatthere are ( n −
1) + ( n −
2) + · · · + 2 + 1 = (cid:0) n (cid:1) = n ( n − total such set partitions.Further we have observed that set partitions with n − n − n ] with n − n − n − i -th block correspond to Bell-permutations of the second kind σ = σ (1) σ (2) . . . σ ( n ) with n − σ ( n ) = i . 14 . A bijection between Bell permutations of the first and the second kind In this part we present a bijection between the set BP ( n ) of Bell permutations over [ n ] intro-duced by M. Poneti and V. Vajnovszki ([Po-Va]) (which we will call Bell permutations of thefirst kind ) and the set BP ( n ) of Bell permutations of the second kind over [ n ].First, we recall the definition of Bell permutations of the first kind.Let π = B /B / . . . /B k ∈ P ( n ) a set partition in its standard representation and let µ : P ( n ) (cid:55)→ BP ( n ), where the permutation µ ( π ) is constructed as follows : • reorder all integers in each block B i in decreasing order; • transform each of these blocks into a cycle. Example 5.1.
Let π = 1279 / / . Then µ ( π ) = σ = (9721)(653)(84) . Remark 5.1.
By definition of Bell permutations of the first kind and by definition of inomcode, if σ ∈ BP ( n ) and f = ˜ φ − ( σ ) is its inom code, then for all i ∈ [ n ] , f ( i ) = inom ( i ) = minimum of the block containing i. Recall also that by definition of Bell permutations of the second kind, if σ ∈ BP ( n ) and f = ˜ φ − ( σ ) is its inom code, then for all i ∈ [ n ] , f ( i ) = inom ( i ) = number of the block containing i. Let us define a map β : BP ( n ) (cid:55)→ BP ( n ).Let σ = C C . . . C k ∈ BP ( n ) be a permutation of BP ( n ) written as a product of disjointcycles, where the cycles have been ordered with their respective minima increasing. Then σ = β ( σ ) is constructed from σ according to the rule :for i = k, . . . ,
2, if the integer i is not in the cycle C i , then insert the cycle C i after i in thecycle containing i . Example 5.2.
Let σ = (9721)(653)(84) . Then σ is obtained as : σ = (9721)(653)(84) −→ (9721)(65384) −→ (972653841) = σ We start by proving that β ( BP ( n )) ⊆ BP ( n ).15 emma 5.1. Let σ ∈ BP ( n ) with k cycles. If σ = β ( σ ) , then w- Exc ( σ ) = { , , . . . , k } .Proof. Let σ = C C . . . C k , where k is the number of cycles of σ .The operation obviously constructs a weak exceedance at each p ≤ k , because it inserts after p a sequence of integers all larger than p . It remains to be proved that if p > k then p is a strictanti-exceedance for σ , that is σ ( p ) < p .If p > k and p is a strict anti-exceedance for σ , that is if σ ( p ) < p , then the constructionnever inserts anything between p and σ ( p ), therefore σ ( p ) = σ ( p ) < p .Let then p > k and σ ( p ) ≥ p . Note that this can only happen if p is the minimum of its cycleof σ , say C t .Let σ ( p ) . . . p be the sequence of elements of the cycle C t of σ , where p = min ( C t ) and σ ( p ) = max ( C t ). Then t ≤ k < p and t is not in the cycle C t because p is the minimum in C t ,hence there is some integer t < t such that t ∈ C t .For i = k, . . . , t + 1 the procedure has not yet modified C t = ( σ ( p ) . . . p ) since all of its elementsare greater than k .Now for i = t , the operation inserts σ ( p ) . . . p in to C t after t (the following diagram showsthis). . . . ( . . . t C t (cid:122) (cid:125)(cid:124) (cid:123) σ ( p ) . . . p . . . ) (cid:124) (cid:123)(cid:122) (cid:125) C t . . . If t (cid:54) = min ( C t ), then t is a strict anti-exceedance for σ , that is, σ ( t ) < t . The steps ofthe transformation for k, k − , . . . , t + 1 have not inserted any integer between t and σ ( t ).When we insert C t in between these two integers, σ ( t ) become the image of p , then we have σ ( p ) = σ ( t ) < t .Suppose t = min ( C t ), then t / ∈ C t for otherwise t ≥ t and hence a contradiction.Thus, there is some integer t < t such that t ∈ C t .For i = t , the operation inserts the sequence of the elements of C t in to C t after t . . . . ( . . . t C t (cid:122) (cid:125)(cid:124) (cid:123) . . . t σ ( p ) . . . p . . . ) (cid:124) (cid:123)(cid:122) (cid:125) C t . . . By the same argument as above, if t (cid:54) = min ( C t ), then we have σ ( p ) < t .Suppose t = min ( C t ), then t / ∈ C t for otherwise t ≥ t and hence a contradiction.Thus, repeating the same procedure a finite number of times we must eventually find someinteger t s such that 1 ≤ t s < t s − (cid:54) = min ( C t s ) and t s ∈ C t s . Then σ ( p ) < t s − .Therefore, p is not a weak exceedance of σ . Hence w- Exc ( σ ) = [ k ].16 emma 5.2. For all σ ∈ BP ( n ) , β ( σ ) = σ ∈ BP ( n ) .Proof. Let σ = C C . . . C k . By Lemma 5.1, we have w- Exc ( σ ) = [ k ] for a certain integer k .Then w- ExcL ( σ ) = (cid:104) α , . . . , α k (cid:105) , where α i = σ ( i ), i = 1 , , . . . , k .Let Seq ( σ ) = (cid:104) γ , . . . , γ k (cid:105) . Then note that for all i ∈ [ k ], the integers γ i and α i , are respectivelythe minimum and maximum integers having inom equal to i . Thus γ i ≤ α i for all i .Therefore, by Theorem 3.1, σ ∈ BP ( n ). Theorem 5.1.
The map β is a bijection between BP ( n ) and BP ( n ) .Proof. We deduce the result from the fact that the following diagram is commutative where : • ˜ φ denotes the inom code; • τ denotes the bijection associating each partition π with its canonical form; • µ denotes the bijection introduced by Poneti and Vajnovszki; • ν denotes the transformation that normalizes any f ∈ ˜ φ − ( BP ( n )) via an order-preservingbijection of Im ( f ) into [ IM A ( f )]; • ζ denotes the transformation that replaces every integer in any f = f f . . . f n ∈ ˜ φ − ( BP ( n ))with the leftmost position where such integer occurs. Figure 2
From the remark 5.1 it is easy to see that the inom code of the permutation β ( σ ) is obtainedby applying ν to the inom code of σ and that the inom code of σ is obtained by applying ζ to the inom code of β ( σ ), (the maps µ and ζ are the inverse of each other when restricted to˜ φ − ( BP ( n )) and ˜ φ − ( BP ( n )) respectively).So we have ˜ φ − ◦ β = ν ◦ ˜ φ − , or equivalently β = ˜ φ ◦ ν ◦ ˜ φ − and therefore β is a bijection.17e can also define directly the inverse of β , a map ϑ : BP ( n ) (cid:55)→ BP ( n ) such that ϑ ( σ ) = σ ,where σ is obtained as follows.Take σ ∈ BP ( n ) and let C C . . . C t be its cycle decomposition. Recall we showed that theset of weak-exceedances of a Bell permutation of the second kind is an interval [ k ].For i = 2 , . . . , k , if i is not the minimum of its own cycle C j , then form a new cycle by takingout of C j the longest sequence of integers greater than i starting immediately after i . Example 5.3.
Let σ = 468912357 = ( in cycle notation and with the weakexceedances in bold. Then σ is obtained as : σ = (1497358)( −→ (1497 )(26)(58) −→ (1 −→ (143)(26)( )(97)(8) = σ Lemma 5.3.
For all σ ∈ BP ( n ) , ϑ ( σ ) = σ ∈ BP ( n ) .Proof. The construction produces a permutation with k cycles in which elements of each cycleare decreasing. Proposition 5.1.
The map ϑ is the inverse of β .Proof. Let σ ∈ BP ( n ), w- Exc ( σ ) = [ k ] and let β ( ϑ ( σ )) = β ( σ ) = σ (cid:48) . Then we show that σ = σ (cid:48) . Since σ has k cycles, σ (cid:48) also has weak-exceedances at { , , . . . , k } , as well as σ .If i > k , then σ ( i ) < i and therefore the construction never changes the image of i and wehave σ ( i ) = σ ( i ) = σ (cid:48) ( i ).Assume that σ ( i ) (cid:54) = σ (cid:48) ( i ) for some i ∈ [ k ].Note that if ϑ ( σ ) = σ , then for every weak exceedance i of σ , one has that σ ( i ) is equal tothe maximum element of the i -th cycle of σ . On the other hand, if σ (cid:48) = β ( σ ) then for everyweak exceedance i of σ (cid:48) , one also has that σ (cid:48) ( i ) is equal to the maximum element of the i -thcycle of σ . This is a contradiction.Therefore σ = σ (cid:48) and hence ϑ = β − . Remark 5.2.
Under the above bijection β : σ (cid:55)→ σ , the number of cycles of σ is equal to thenumber of weak exceedances of σ . cknowledgements Both authors are members of the project CoRS (Combinatorial Research Studio), supported bythe Swedish government agency SIDA. The most significant advances of this research work havebeen made during two visits of the first author to IRIF. The first visit was entirely supported byIRIF, the second visit was substantially supported by ISP (International Science Programme) ofUppsala University (Sweden) and partially supported by IRIF. The authors are deeply gratefulto these two institutions. We also thank our colleagues from CoRS for valuable discussions andcomments.
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