Extensions of set partitions and permutations

Extensions of a set partition obtained by imposing bounds on the size of the parts is examined. Arithmetical and combinatorial properties of these sequences are established. Mathematics Subject Classifications: Primary 11B83; Secondary 11B73, 05A15, 05A19.


Introduction
A partition of a set [n] := {1, 2, . . ., n} is a collection of nonempty disjoint subsets, called blocks, whose union is [n].The Stirling numbers of the second kind n k count the number of partitions of [n] into k non-empty blocks.The total number of partitions of [n] is given by the Bell number starting with 1, 1, 2, 5, 15, 52, 203.In this count, the order of the blocks is not relevant.On the other hand, if the order of the blocks is important, then the total number of the electronic journal of combinatorics 26(2) (2019), #P2.20 partitions of [n] is known as the Fubini number F n .The expression is the analogue of (1), starting at 1, 1, 3, 13, 75, 541, 4683.Additional information about set partitions may be found in [34].Similar sequences of numbers are obtained by enumerating permutations on n elements with k cycles.For example, the (unsigned) Stirling numbers of the first kind, denoted by n k .The Stirling numbers of both kinds are related by the relation where δ n,k is the Kronecker delta symbol.
The literature contains (at least) two generalizations of these combinatorial sequences.As an example, for r ∈ N, an r-partition of n is one in which the first r elements are in distinct blocks.The r-Stirling numbers, denoted by n k r , count the r-partitions of [n + r] into k + r blocks and satisfy the recurrence , for n ∈ N, 0 k n and 1 r k, (4) These numbers were introduced by Broder [7].The r-Stirling numbers may be expressed in terms of the classical Stirling numbers by Mező [37] introduced the r-Bell numbers by with B n,0 = B n , the Bell numbers in (1).These numbers satisfy the recurrence Similarly, an r-permutation, is one in which the first r elements are in distinct cycles.The number of all r-permutations of [n + r] into k + r cycles are counted by the r-Stirling numbers of the first kind, denoted by n k r .Other combinatorial objects introduced in this manner include the r-derangement numbers [60,59], the r-Bell numbers [37], the r-Whitney numbers and their q-analogues [8,35,36,40,42,51], the r-Lah and r-Lah-Whitney numbers [46,52,54], the r-Fubini and r-Whitney-Fubini numbers [14] and the r-Whitney-Eulerian numbers [50,53].The extension of the results presented here for these other classes is the subject of current work.These generalizations are known as incomplete combinatorial structures.They come from imposing a restriction on the size of the blocks and cycles.If the size of the substructure (block, cycles, etc.) is required to be bounded from above, then one speaks of a restricted combinatorial structure; the case of a lower bound is named an associated combinatorial structure.In the situation where the notion of special elements are included, the letter r is added to the name.
For the convenience of the reader, this section contains the list of the numbers discussed in the present work.

A partition of [n] is a collection of non-empty subsets, called blocks, whose union is
[n].
3. An r-partition of n is a partition of [n] in which the first r elements are placed in distinct blocks.The numbers 1, 2, . . ., r are called special.A special block is one containing a special element.
4. The Stirling number of the second kind n k counts the number of partitions of [n] into k non-empty subsets (or blocks). 5.The Bell number B n counts the total number of partitions of [n] into non-empty subsets.
6.The Fubini number F n counts the total number of partitions of [n] into non-empty blocks, where the order in which the blocks appear is taken into consideration.
7. The r-Stirling numbers of the second kind n k r count the number of r-partitions of [n + r] into k + r blocks.the electronic journal of combinatorics 26(2) (2019), #P2.20 13.The associated r-Stirling numbers n k m,r count the number of r-partitions of [n+r] into k + r subsets with blocks of size at least m.
14.The restricted Bell numbers B n, m count the total number of partitions of [n] into non-empty subsets with blocks of size at most m.
15.The restricted Fubini numbers F n, m count the total number of ordered partitions of [n] into blocks of length at most m.
16.The associated Fubini numbers F n, m count the total number of ordered partitions of [n] into blocks of length at least m.
17.The restricted r-Bell number B n, m,r is the number of r-partitions of n, with block size at most m.
18.The associated r-Bell number B n, m,r is the number of r-partitions of n, with block size at least m.
The work presented here contains combinatorial and arithmetical information on these sequences of numbers.The arithmetical part includes congruences as well as valuations.Recall that, for a prime p and n ∈ N, the p-adic valuation of n is the highest power of p that divides n.An important tool in the analysis of valuations is Legendre's formula [30] for the valuation of factorials: where s p (n) is the sum of digits of n in its base p representation.
Remark 1.It is often the case that an analytic expression for ν p (a n ) is hard to find.In many situations one finds that the valuations are given by a valuation tree.This concept is illustrated with ν 2 (n).The vertices of the tree have associated to them a subset of indices, some of these vertices have descendants one level down.The rules are these: start with a root vertex and associate to it the set N. Since the valuation {ν 2 (n) : n ∈ N} is not constant, this vertex is split onto two new vertices (one per residue class modulo 2), which form the next level.To the first vertex one associates the indices {n ∈ N : n ≡ 0 (mod 2)} and to the second one {n ∈ N : n ≡ 1 (mod 2)}.Since the valuation ν 2 (n) of indices associated to the second vertex is constant (≡ 1), this vertex is declared terminal.
The constant value 0 is then attached to this vertex.The first vertex has non-constant valuation, so its indices are split according to its residues modulo 4, into {n ≡ 2 (mod 4)} and {n ≡ 0 (mod 4)}.The process continues: the first vertex has constant valuation 1 and the second is then split modulo 2 3 to continue the process.In the situation where this process does not terminate in a finite number of steps, one says that the valuation admits a tree structure.
Another question of interest in the current work is to examine the periodicity of sequences modulo a number m ∈ N. Definition 2. An integer sequence A = (a n ) n 0 is a periodic sequence modulo m, with period t, if there exists s 0 such that a n+t ≡ a n (mod m), for n s.The smallest t is called the minimum period of A.
Remark 3. Let p be a prime and {a n } an integer sequence of period p.If a j ≡ 0 (mod p) for 1 j p, then ν p (a n ) ≡ 0.
Example 4. A direct application of the pigeon-hole principle shows that the Fibonacci numbers, defined by the recurrence f n = f n−1 + f n−2 and initial conditions f 1 = f 2 = 1, is a periodic sequence modulo m, for any m ∈ N. The minimal period for m = 5 is 20.
Among the other questions considered here is the distribution of the last digit of a sequence.The periodicity of the last digit has been studied for several combinatorial sequences.For example, the last digit of the Fibonacci numbers is a periodic sequence of period 60; see [58,61] for more information.
The plan of the paper is as follows: Section 2 presents properties of the Bell numbers B n which will be extended to other families of numbers.Section 3 considers the restricted Bell numbers B n, m which count the number of partition of [n] with blocks of size at most m.Section 4 extends the results of the previous section to r-partitions of [n + r] with blocks of length at most m.Recurrences, congruences and divisibility issues are discussed.The distribution of the last digit of B n, 2,r and B n, 3,r is settled in Section 5.The general problem for r 4 remains to be determined.The valuations ν 2 (B n, m,r ) are completely determined for m = 2, 3 in Section 6.The general case remains open.Similar results for the associated Bell numbers B n, m,r are presented in Section 7. The question of divisibility of these numbers remains an open question.An extension of these numbers to polynomials is discussed in Section 10.Exponential generating functions for these polynomials are established.Section 8 discusses the restricted Stirling numbers of the second kind and Section 9 presents a combinatorial proof of an identity involving these numbers.The Fubini numbers, counting partitions of [n] taking into account the order of the participating blocks are discussed in Section 11.Recursions are established as well as the periodicity of the last digit.The divisibility question is presented in detail for the primes 2 and 3.For primes p 5 experimental results are discussed.The restricted and associated Fubini numbers F n, m and F n, m are discussed in Section 12. Their arithmetic properties appear in Section 13.These results are extended to the r-Fubini numbers in Section 14. Finally a generalization of factorials is discussed in Section 15.
Hall [23] showed that {B n (mod p)} has period N p = p p −1 p−1 .This result was rediscovered by Williams [63], who showed this is the minimum period when p = 2, 3 and 5. Radoux [49] conjectured that N p is always the minimum period.Several authors have established special cases.For example, Montgomery et al. [45] proved it for most primes p < 180.
The valuation of B n are discussed in Amdeberhan et al. [1].For the prime p = 2, it is shown that with initial condition B 0, m = 1 in analogy to (9).Some congruences appear in [43].For p prime and 0 m < p, This shows that B n, p (mod p) is a periodic sequence.The particular value B p, m ≡ 1 (mod p) follows from B 0, m = 1.The explicit expression presented in [43] shows that generalizing (14).
Remark 6.The periodicity of B n, m (mod p) shows that the last digit of B n, m is also a periodic sequence.For example, for m = 5, the sequence B n, 5 (mod 2) has period 8 and B n, 5 (mod 5) has period 20.Therefore the last digit of B n, 5 has period lcm{8, 20} = 40.
Some analysis of the p-adic valuation of the restricted Bell numbers appears in the literature.Amdeberhan et al. [4] established an expression for the 2-adic valuation of the restricted Bell numbers B n, 2 : The sequence B n, 2 coincides with the number of involutions of n elements, denoted in [4] by Inv 1 (n).This sequence is also called Bessel numbers of the second kind, see [10] for further information.
The valuation for the prime p = 3 is easy to determine using Remark 3.
Proof.Formula (14) shows that B n, 2 is periodic modulo 3. The result follows from the values B 2, 2 = 1, B Start with a root node representing all positive integers.Then observe that ν 5 (B n, 2 ) = 0 if n ≡ 4 (mod 5).Construct a new level, called the first level, with 5 nodes connected to the root and label them by the residues classes modulo 5.Each node corresponds to a collection of indices.It is denoted by V 1,j and it corresponds to the indices The vertex is called terminal if the valuation ν 5 (B m, 2 ) for every m ∈ V 1,j is independent of m.For example, since ν 5 (B 5n, 2 ) = 0, the vertex V 1,0 is terminal.The constant valuation of a terminal vertex V is called the valuation of the vertex and is denoted by ν p (V ), or ν p (V ; B n, 2 ), to mention the sequence under study.For example, ν 5 (V 1,0 ; B n, 2 ) = 0.In this example, there are four terminal vertices V 1,j : j = 0, 1, 2, 3.Each of them has valuation 0. Now construct the second level by splitting the indices in V 1,4 modulo 5 2 .This gives the vertices On this level, there are four terminal vertices, with valuation This represents the fact that ν 5 (B 5n+4, 2 ) = 1 for n ≡ 4 (mod 5).Repeating this process and now forming the third level gives It is conjectured that this process can be continued indefinitely.The resulting tree is called the valuation tree for the prime 5 and the sequence B n, 2 ; or simply, that ν 5 (B n, 2 ) has a valuation tree structure.

Restricted r-Bell Numbers
This section introduces a new extension of the restricted Bell numbers B n, m studied in Section 3. Some basic properties of arithmetical and combinatorial properties are presented.The definition employs the notion of r-partition given in (6) of Section 1.
Definition 9.For n, m, r ∈ N, the numbers B n, m,r count all r-partitions of [n + r] such that each block has size at most m.These numbers are called the restricted r-numbers.
The elements 1, 2, . . ., r will be called special elements and a block of a partition is called special if it contains a special element.
Remark 10.The case r = 0 yields the restricted Bell numbers of Section 3 and the limiting case gives the r-Bell numbers in (6).A second special case is B n, 1,r = B n, 1 = 1, since the size of each block must be exactly 1 and then the condition on special block is vacuous.
The first result gives a recurrence similar to (13).Observe that in (20), the index r on the left-hand side is reduced by 1 on the right.Iteration of this recurrence takes B n, m,r to B n, m,0 = B n, m .This is computable using (13).
Theorem 12.The restricted r-Bell numbers, B n, m,r , satisfy the recurrence for n 1, r 1 and m 1.The initial values are B n, m,0 = B n, m from (13) and B 0, m,r = 1 and B n, m,r = 0 for r < 0 or n < 0.
Proof.Suppose the first special block is of size , where 1 m.This block contains the minimal element 1, and the rest of the block is formed by choosing − 1 elements, with 0 − 1 m − 1.Therefore, the number of r-partitions of [n + r] with exactly elements in the first block is n B n− , m,r−1 for 0 m − 1. Summing over completes the proof.
A second recurrence is presented next.
Theorem 13.The restricted r-Bell numbers, B n, m,r , satisfy the recurrence for n 1, r 0 and m 1.The initial values are the same as in Theorem 12.
the electronic journal of combinatorics 26(2) (2019), #P2.20 Proof.Let i be the size of the block containing the last element, namely n + r.Then 1 i m.If this block is special, there are r ways to choose the special element in this block.The remaining non-special elements, with 0 m − 2, can be chosen in n−1 ways.This corresponds to the first sum in (21).The second sum appears for the non-special elements.
The recurrence relations are now used to establish some congruences of restricted r-Bell numbers.The proof uses an elementary congruence established below.Lemma 14.Let n, k, p, s be non-negative integers and p a prime with 0 k < p.Then, for each s ∈ N, Proof.The binomial theorem gives On the other hand, if The congruence now comes by matching the corresponding coefficients.
The next result is preliminary for a further generalization of ( 14) stated in Theorem 16.
Lemma 15.Let p be a prime, s ∈ N and m < p. Then B p s , m,r ≡ 1 (mod p s ).

Proof. Theorem 12 gives
Iteration and ( 15) produce the result.
The next statement establishes the periodicity of the restricted Bell numbers modulo a power of a prime.
Theorem 16.Let n, r, s ∈ N, p a prime and 1 m < p.As a function of n ∈ N, the restricted r-Bell numbers, B n, m,r , is a periodic sequence modulo p s , with period p s .That is B n+p s , m,r ≡ B n, m,r (mod p s ).
Proof.Proceed by induction on n.Lemma 15 gives n = 0. Theorem 13 and Lemma 14 give The proof is complete.
Remark 17.The sequence {B n, m,r } (mod p) is periodic modulo p.The result of Hall [23] for the periods of the Bell numbers modulo p cited in Section 2 has been extended by Mező and Ramírez [41] to the r-Bell numbers.
5 The last digit of the restricted r-Bell numbers.
Given x ∈ N, the value x (mod 10) is the last digit of x.This section discusses properties of the last digit of the restricted r-Bell numbers, B n, r,m .The proofs use the congruence in Theorem 16.
The discussion starts with {B n, 2,r } for r = 1, 2, 3. Figure 1 shows the first 100 values of the last digit in {B n, 2,1 }.The data suggests that this is a periodic sequence of period 5. Theorem 18. Fix r ∈ N. The last digit of the sequence {B n, 2,r } is a periodic sequence of period 5; that is, for any n ∈ N B n+5, 2,r ≡ B n, 2,r (mod 10) for n 2.
The same statement holds for {B n, 3,r }, for n 4.
Proof.Theorem 16 with p = 5 gives B n+5, 2,r ≡ B n, 2,r (mod 5).Theorem 13 and the corresponding initial values, show that B n, 2,r is an even number.Therefore the congruence extends from modulo 5 to modulo 10.The proof for B n, 3,r is similar.

p-adic valuations of restricted r-Bell numbers
Given a sequence {a n } of positive integers and a prime p, the sequence {ν p (a n )} of p-adic valuations offers interesting challenges.Interesting examples include the 2-adic valuation of Stirling numbers of the second kind [3,15,24,31], the 2-adic valuation of a sequence of integers appearing in the evaluation of a definite integral [33] and also the 2-adic valuation of domino tilings [11].
Amdeberhan et al. [4] established an expression for the 2-adic valuation of the restricted Bell numbers B n, 2 : The sequence B n, 2 coincides with the number of involutions of n elements, denoted in [4] by Inv 1 (n).This sequence is also called Bessel numbers of the second kind, see [10] for further information.The 2-adic valuation of the restricted r-Bell numbers B n, 2,r follows a similar pattern.Figure 3 shows the first few values of ν 2 (B n, 2,6 ).Jung et al. [26] described ν 2 (B n, 2,r ).The general formula is divided into many cases.For example, if r ≡ 0 (mod 4), then where α k + 2. The reader is invited to verify that, in the case n ≡ 3 (mod 4), the valuation ν 2 (B n, 2,r ) admits a simple formula for r ≡ 0 (mod 8) and it is more complicated if r ≡ 4 (mod 8).
The next goal is to discuss the 3-adic valuation ν 3 (B n, 2,r ).The first results give some congruences for the restricted r-Bell numbers modulo 3.
The next lemma extends the analysis of Lemma 22 and its corollary to indices modulo 9.The proof is left to the reader.
It remains to analyze the sequence ν 3 (B n, 2,r ) for n ≡ 8 (mod 9).The description below, for r = 8, describes the valuation tree, as introduced in Remark 1.
Remark 25.The description of the valuation ν 3 (B n, 2,8 ) is given by a valuation tree.The root of the tree corresponds to all indices n ∈ N. A sequence of nodes is constructed as follow: each node has attached a collection of indices.In this construction, one asks the following question: is the valuation ν 3 (B n, 2,8 ) independent of n.If the answer is positive, then the vertex is declared terminal and the constant value is assigned to it.If the answer is negative, the index set is split according to the residue modulo a power of the prime p = 3.
The complete analysis of the valuations ν p (B n, m,r ) cannot (up to now ) be derived from this type of arguments.

Associated r-Bell Numbers
This section discusses some properties for the associated r-Bell numbers B n, m,r which count the number of r-partitions of [n + r] with blocks of length at least m.The results are similar to those presented in the previous sections for the restricted r-Bell numbers B n, m,r .The first statement is the analog of Theorems 12 and 13.The proofs are similar, so they are not presented here.
Theorem 27.For n 1, r 0 and m 1 the associated r-Bell numbers B n, m,r satisfy the recurrences with the initial values B 0, m,0 = 1 and B n, m,0 = 0 for 0 < n m − 1, and for r > 0 and 0 n < (m − 1)r the values are B 0, m,0 = 0. Additionally, for r 1, The next statement offers a relation between the restricted r-Bell numbers B n, m,r and the associated r-Bell numbers B n, m,r .
Theorem 28.The associated r-Bell numbers B n, m,r can be calculated from the r-Bell numbers B n,r and the restricted r-Bell numbers B n, m,r , via Proof.Let B n,r denote the set of all r-partitions of [n + r], with cardinality B n,r .Suppose This is the set of all non-special elements appearing in blocks of length less than m.Let S i and Summing over i yields the desired result.
the electronic journal of combinatorics 26(2) (2019), #P2.20 The previous result is now considered modulo a prime p.
Corollary 29.Let p be a prime.Then Moreover, B p, m ≡ 1 (mod p).
Proof.Theorem 28 gives Theorem 3 of [41] gives B p,r ≡ r + 2 (mod p).This implies the first equality.The second identity follows from Theorem 16.The last congruence is the special case r = 0.
Remark 30.The congruence in Corollary 29 may be written as This form is useful in an inductive argument (in r) of modular properties of B p, m,r .For example, r = 1 yields Remark 31.Unlike the restricted r-Bell numbers, the associated r-Bell numbers do not have a predictable behavior for their last digit.This unpredictability extends to their valuations.Moll et al. [44] studied the function ν 2 (B n, 2 ), proving that For n ≡ 1 (mod 3), the valuation satisfies ν 2 (B n, 2 ) 1.A more detailed study of this function is in progress.

A combinatorial identity
The Stirling numbers n k count the number of partitions of [n] into k non-empty blocks.It is natural to consider the extensions n k r of r-partitions of [n + r] into k + r non-empty blocks as well as n k m,r , the number of r-partitions of [n + r] into k + r non-empty blocks of size at most m.There are also families of corresponding associated numbers.These sequences are discussed in [44].Further information about these numbers may be found in [6,9,28].
The goal of this section is to present a combinatorial proof of an interesting identity for n k 2,r given in [26].This sequence was studied by Cheon et al. [9] by means of Riordan arrays.
Theorem 32.The restricted r-Stirling numbers and the associated r-Stirling numbers satisfy the following recurrences The notation [n]   k is used for the set of partitions of [n] into k non-empty blocks with Theorem 33.For n, r 0, the identity • i ∈ π i for i ∈ [r], i.e., π i is a signed special block containing their indices.
• For i r, the blocks contain only positive numbers.
i.e., the elements on the signed blocks, giving the disjoint union In the set R π we have put the elements for the first r blocks, and so counting elements in the k remaining blocks is equivalent to partition [n − i] into k blocks and attaching a sign to them, except for the minimal ones.This yields The right-hand side of the required identity appears as the cardinality of A n,k,r .
In order to complete the proof, consider the map For example, take n = 10, j = 6, r = 2, k = 2 and This map is a bijection and the identity follows.Details are left to the reader.
Remark 34.In the case r = 0, the statement above gives a relation for the Bessel numbers of the second kind (see [64]): Remark 35.Cheon et al. [9] studied a related sequence b r (n, k) orthogonal to n k 2,r , i.e., The numbers b r (n, k) := (−1) n−k b r (n, k) are called the unsigned r-Bessel numbers of the first kind, with exponential generating function An interesting combinatorial interpretation of this sequence, using the concept of Gpartitions, is given in [9].

Generalized Howard's Identities
The restricted Stirling numbers of the second kind, n k m and associated Stirling numbers of the second kind, n k m were introduced in Section 8.The goal here is to present a combinatorial proof of some identities given by Howard in [25].The symbol [n]   k denotes the set of n-combinations of k-elements, with cardinality n k .Theorem 36.Let n ∈ N and 0 k n.Then the electronic journal of combinatorics 26(2) (2019), #P2.20 Proof.Elementary manipulations transform the desired identity to Define the function These polynomials extend the Bell numbers B n , since B n (1) = B n .Mező [37] introduced the r-Bell polynomials by where n k r are the r-Stirling numbers of the second kind.Further generalizations of these polynomials appear in Corcino et al. [14].The restricted (associated) r-Bell numbers appear as The objects of interest in this section are two families of polynomials: the restricted r-Bell polynomials B n, m,r (x) and the associated r-Bell polynomials B n, m,r (x) defined by The exponential generating function of these families appeared in [6,9].B n, m,r (x) The classical Stirling numbers of the second kind satisfy the identity where x k is the falling factorial defined by ) for k 1 and x 0 = 1.A generalization for the incomplete r-Stirling numbers is presented next, the relation to r-Bell polynomials is stated in Theorem 40.The proof is a direct application of Theorem 8 of [42].
Proof.Theorem 39 implies The proof is complete.
Example 41.The identity produces the expression The analogous result the associated type is stated next.
Theorem 42.The associated r-Bell polynomials satisfy the identity x ! .

Fubini Numbers
The Stirling numbers of the second kind n k count the number of partitions of [n] into k non-empty blocks.The total number of partitions is given by the Bell number B n in (1).The corresponding counting situation, where now the k blocks are ordered, is given by the Fubini numbers F n , also called the ordered Bell numbers.They are given by as stated in ( 2).This section discusses some of their properties.
Remark 43.The explicit formula for the Stirling number of the second kind gives Remark 44.The Fubini numbers satisfy the recurrence with initial condition F 0 = 1.A proof is given in Corollary 72.
Remark 45.The exponential generating function for {F n } is given by The next statements deal with modular properties of the Fubini numbers.
Theorem 46.Let p be a prime.Then {F n (mod p)} is a periodic sequence of period p − 1.
Proof.Fermat's little theorem gives a p ≡ a (mod p) and the identity i=0 the electronic journal of combinatorics 26(2) (2019), #P2.20 imply that A similar argument shows that {F n } is periodic modulo p s , with period p s−1 (p − 1), i.e., see Barsky [5].Diagana and Maïga [16] established the generalization with gcd(t, p) = 1.For p prime, Good [20] conjectured that p − 1 is the minimal period of the Fubini numbers modulo p.This conjecture was verified in [20], for 2 p 73.The general case was established in Poonen [48].
Theorem 47.Let p be a prime and s ∈ N. Then {F n } modulo p s is a periodic sequence, with minimal period p s−1 (p − 1).
The expression (29) yields the following result.
The next result establishes the structure of the last digit of F n .The proof uses the periodicity of the F n for p = 2, 5, first established by Gross [22].
The recurrences stated below were proven first by Poonen [48] using induction.Diagana and Maïga [16] used the Laplace transform of a p-adic measure to present a new proof.Two different proofs are presented below: one using combinatorial identities and then a bijective proof.
Theorem 50.Let n, q ∈ N. Then Proof.The identity (see [13, pp. 228]) produces Multiplying by 2 q to obtain the result.The second identity is proven in a similar manner.
Claim: ϕ is a bijection.To show this, define ).
Step 2. ϕ • ψ = Id.This time, the composition is . Also, Y = |λ| i=1 λ i by definition of partition and the projection sends λ to λ.
This shows that ϕ is a bijection.It follows that For the second part of the identity, consider the function contains the range of the function, defined as the number of parts of the resulting partition.The map ϕ is a bijection and its inverse is given by As an example, consider X = {2, 4, 5, 7, 8, 9, 11} ⊆ [20] and the electronic journal of combinatorics 26(2) (2019), #P2.20 Combining ( 35) and ( 36), gives the stated identity.
Diagana and Maïga [16] used Theorem 50 to establish some interesting congruences for the Fubini numbers.
In particular, if n = q = p is a prime number, then Corollary 52.Let q ∈ N. Then Velleman and Call [57] gave a combinatorial identity for F n , similar to (29).An alternative proof is presented next.
Theorem 53.For all n 1, Proof.The Eulerian numbers A n,k , given by count the number of permutations containing exactly k runs.Let A n,k be the set of these permutations.Information about these permutation appear in [21].
The identity (37) and prove the result.The proof of ( 38) is presented next.
Therefore the partition is being encoded as a permutation where the blocks start with a descent in the permutation or where we indicate with the resulting set which has index of ascents in the permutation.This function has the natural inverse: given a permutation with k − 1 descents and given a subset of the ascents, we can generate the partition in the following way, consider π = 243581769 ∈ A 9,4 where the descents are underlined.As the first element of the permutation does not count as ascent, then we have 5 = |{2, 4, 5, 7, 9}|, where the set is the indices of the ascents in π, places to choose for creating a new block.If we choose X = {4} then we can construct the partition ({2, 4}, {3}, {5, 8}, {1, 7}, {6, 9}).
12 Restricted and associated Fubini Numbers The Fubini numbers F n count the number of ordered set partitions.It is natural to generalize them by restricting the size of the blocks used in the partitions.This gives the restricted Fubini numbers, F n, m , where the blocks are of size at most m and the associated Fubini numbers F n, m , with blocks of size at least m.In terms of the restricted/associated Stirling numbers, the identities Theorem 39 provides the following relations: the electronic journal of combinatorics 26(2) (2019), #P2.20 where In the case m n, this identity yields the classical formula An elementary argument shows that this counts the total number of functions from [n] to the set [t].The equations ( 39) and ( 40) admit the following combinatorial interpretation: the expression f Remark 54.Komatsu and Ramírez [29] found the exponential generating functions for the restricted/associated Fubini numbers: The next statement is analogous to the identity (34).
Theorem 55.For n 0 we have Proof.The expressions in (39) and (41) give The result follows by comparing coefficients.The proof of the second identity is similar.
Theorem 56.Let n, q ∈ N. Then Proof.Theorem 55 and (39) imply The second identity follows in a similar manner.
The previous theorem is now used to generate some congruences.
Corollary 57.Let q, n ∈ N. Then The next statement is a generalization of Theorem 53.
the electronic journal of combinatorics 26(2) (2019), #P2.20 Theorem 58.The restricted/associated Fubini numbers satisfy the identities Proof.The proof is analogous to the one given for Theorem 53.Simply use the combinatorial observation (39).
The final result in this section is an identity relating the associated Fubini numbers F n, k with the Fubini numbers F n and the incomplete Stirling numbers i j k and i j k .
Theorem 59.Let n ∈ N. Then Proof.Let π = (π 1 , . . ., π ) ∈ F n be an ordered partition.Write [n] = A π ∪ B π where The sets S i and it follows that The identity follows from here.
13 Arithmetical properties of the restricted/associated Fubini numbers This section discusses some arithmetical properties of the Fubini numbers F n and their generalizations.Particular emphasis is placed on congruences and p-adic valuations.
Proof.Proceed by induction and use the recurrence stated in Remark 45, to obtain the electronic journal of combinatorics 26(2) (2019), #P2.20 For example, if n ≡ 1 (mod 4), it turns out that ν 5 (F n ) = 0; that is, ν 5 (F 4n+1 ) = 0 for all n ∈ N. Similarly, ν 5 (F 4n+2 ) = 0 and ν 5 (F 4n+3 ) = 0 for all n ∈ N. (It is a coincidence that each of the classes has the same value, namely 0. The important point is that this value is the same for each index in the class).In the case n ≡ 0 (mod 4), the value ν 5 (F 4n ) does depend on the index n.Therefore, based on experience acquired with other sequences, these numbers are split modulo 5 to produce five classes of indices: 20n, 20n + 4, 20n + 8, 20n + 12, and 20n + 16.Now there are four classes for which the valuation of the Fubini number with index in the class, has a valuation independent of the index.For example, ν 5 (F 20n ) = 1 for all n ∈ N. Similarly As before, in the remaining case n ≡ 4 (mod 20), the value ν 5 (F 20n+4 ) does depend on the index n.
Conjecture 64.The process described above continues ad infinitum.At each step, there is a single class where the valuation is not constant.Moreover, this phenomena happens for every prime p 5.
Definition 65.The restricted Fubini numbers F n, m count all the partitions of [n] into blocks of length at most m, where the order in which the blocks appear is taken into consideration.The corresponding associated Fubini numbers F n, m are defined in a similar form, now with blocks of length at least m.
Remark 66. Mező [38] established the recurrence Similarly, there is a recurrence for the associated Fubini numbers as This section discusses some elementary arithmetic properties of the numbers F n, m and F n, m .Lengyel [32] establish some additional arithmetical properties for this sequence.The first result states that, in the case p = 2, these numbers are related to the restricted and associated Stirling numbers.
the electronic journal of combinatorics 26(2) (2019), #P2.20 Proof.Use Corollary 57 with q = 2 and observe that, since n 1 m = 1 for 1 n m and 0 otherwise, then F n, m is odd for 1 n m and even for all n > m.
The 2-adic behavior of the restricted Fubini numbers is discusses next.Figure 4 shows the first few values of the sequence ν 2 ({F n, 2 }).
An analytic expression explaining this figure is presented in the next theorem.
Theorem 68.The 2-adic valuation for the restricted Fubini numbers F n, 2 is where s 2 (n) is the sum of the digits of n in its binary expansion.
Proof.The proof is by induction, and is divided into four cases according to the residue of n modulo 4. The symbols O i denote an odd number.If n = 4k then (52) with m = 2 and the induction hypothesis yields the electronic journal of combinatorics 26(2) (2019), #P2.20 If k is odd, then F 4k, 2 = 2 2k−1−s 2 (2k−2) O 5 , and it follows that . This is a direct consequence of Legendre's formula for the p-adic valuation of factorials: The remaining cases are analyzed in a similar manner.
Symbolic computations produce the next statement.The reader is invited to produce a proof in the style presented for the previous theorem.
Theorem 69.The 3-adic valuation of F n, 2 has a 3-block structure; that is, for n ∈ N: The common value is given by where s 3 (n) is the sum of the digits in the expansion of n in base 3.

r-Fubini Numbers
The r-Fubini numbers, F n,r , have appeared in [39].They are defined as the number of ordered r-partitions of [n + r].Thus, The first statement gives a recurrence for F n,r .The initial condition F 0,r = r! is clear from the definition.
Theorem 71.Let n ∈ N. Then the r-Fubini numbers satisfy the recurrence the electronic journal of combinatorics 26(2) (2019), #P2.20 Proof.Assume the last block in an ordered r-partition of [n + r] is non-special and has k elements, with 1 k n.There are n k ways to choose these elements and the remaining n − k elements can be ordered in F n−k,r ways.If the last block is special then the number of choices is r n k−1 F n−(k−1),r−1 .Summing over k gives The special case r = 0 gives the next result, stated in Remark 44.
Corollary 72.The Fubini numbers satisfy the recurrence Proposition 73.Let p be a prime.Then F p,r ≡ 2r! + rF p,r−1 (mod p).
The next result gives an exponential generating function.x x n n! = e x F r (x) + re x F r−1 (x).In the case where the restriction is that none of the cycles of a permutation of n contain more than m items, one obtains the restricted Stirling numbers of the first kind, denoted by n k m (cf.[18,38,44]).The associated Stirling numbers of the first kind n k m counts the case where the size of the cycles are at least m.
The corresponding r-generalizations, those where the first r elements are in distinct cycles (cf.[6,9,28]), produce the restricted r-Stirling numbers of the first kind n k m,r .The associated Stirling numbers of the first kind n k m and the associated r-Stirling numbers of the first kind n k m,r are defined similarly.The first statement is a combinatorial proof of an identity of Howard [25].
Theorem 78.For k, n ∈ N, the identity Proof.The change of indices j = n − k − , converts the desired identity into The right-hand side corresponds to the decomposition of the permutation of n with k cycles into its fixed points and the cycles of length 2.
The next identity admits a proof similar to the one presented for Theorem 36.As before, some polynomials are constructed with these families of numbers.
Similarly, for t, r ∈ N define w t,r (x) by Then 16 Conclusions A variety of numbers of combinatorial origin are discussed.These sequences are obtained by restricting sizes of substructures in set partitions, either from above or below.Arithmetic properties include congruences and structures of their p-adic valuations were discussed.

8 .
The r-Bell numbers B n,r count the total number of r-partitions of [n + r]. 9.The r-Fubini numbers F n,r count the total number of ordered r-partitions of [n + r].10.The restricted Stirling numbers n k m count the number of partitions of [n] into k subsets with blocks of size at most m.11.The associated Stirling numbers n k m count the number of partitions of [n] into k subsets with blocks of size at least m.12.The restricted r-Stirling numbers n k m,r count the number of r-partitions of [n+r] into k + r subsets with blocks of size at most m.

Figure 2
Figure 2 shows the first 200 values of the last digit of the sequences {B n, 4,0 } = {B n, 4 }.The data suggests that this sequence has period 20.

Figure 2 :
Figure 2: The last digit of the sequences {B n, 4 }.

Theorem 79 . 2 =0( + 1 ) 1 =0(n − 1 )
Let n, k, m ∈ N. of numbers is introduced next.Definition 80. Let A n, m,r the total number of r-permutations of [n + r] with the condition that each cycle has size at most m.This sequence is called restricted r-factorial numbers.The associated r-factorial numbers A n, m,r are defined in a similar manner.The following identities are immediate: Remark 81.In the special case m = 2 one has A n, 2,r = B n 2,r .The numbers A n, 2,r are the r-derangements numbers, discussed in[60,59].The next statements are analogous to results discussed in the previous sections.The details of the proofs are left to the reader.Theorem 82.For n 1, r 0 and m 2 the restricted r-factorial numbers satisfy the recurrence relationA n, m,r = r m−(n − 1) A n−1− , m,r−1 + m−A n−1− , m,r .Moreover, A n, m,r = m−1 =0 n A n− , m,r−1 .Theorem 83.For n 1, r 0 and m 1 the associated r-factorial numbers satisfy the recurrence relationA n, m,r = r n−1 =m−2 ( + 1)(n − 1) A n−1− , m,r−1 + n−1 =m−1(n − 1) A n−1− , m,r .Moreover, A n, m,r = n =m−1 n A n− , m,r−1 .

Definition 84 .r e x z+ z 2 2 += z m− 1 + 2 + • • • + x m m t 1 +
The restricted r-factorial polynomials, A n, m,r (x), and the associated rfactorial polynomials, A n, m,r (x) are defined by the expressionsA n, m,r (x) = n k=0 n k m,r x k and A n, m,r (x) = n k=0 n k m,r x k .The exponential generating functions of these polynomials are stated next.Theorem 85.The exponential generating function of the restricted/associated r-factorial polynomials are∞ n=0 A n, m,r (x) z n n! = 1 + z + z 2 + • • • + z m−1 z m + • • • r e x z m m + z m+1 m+1 +••• .The next statement is the analogous to the Theorem 39.the electronic journal of combinatorics 26(2) (2019), #P2.20Theorem 86.For t, r ∈ N, defineh t,r (x) = 1 + x + x x + x 2 + • • • + x m−1 r .Thend n dx n h t,r(x) = 1 is clear: there is a single way to partition [n] into non-empty blocks with at most one element.On the other hand, if m > n, then B n, m = B n .Miksa et al. established (as Theorem 2) the recurrence Remark 8.The discussion of ν 5 (B n, 2 ) is used now to introduce the concept of a valuation tree.This notion has been mentioned in the context of the valuation ν 3 (B n ) for n ≡ 4, 8, 17, 21 (mod 26), in Section 2. The statements made here are based on computer experiments.The reader is encouraged to use congruences above to provide rigorous proofs.

Table 1 :
The fundamental period for the last digit.