The perimeter and the site-perimeter of set partitions

In this paper, we study the generating function for the number of set partitions of [n] represented as bargraphs according to the perimeter/site-perimeter. In particular, we find explicit formulas for the total perimeter and the total site-perimeter over all set partitions of [n]. Mathematics Subject Classifications: 05A18


Introduction
A partition of a positive integer n is a non-increasing sequence of positive integers, called parts, whose sum is n.A composition is a partition in which the parts may come in any order, as originally defined by MacMahon [11].Compositions can be represented as bargraphs.A bargraph is a column convex polyomino such that the lower edge lies on a horizontal axis, when it is drawn on a regular planar lattice grid and is made up of square cells.Clearly, the number parts and the size of a composition is the number columns and the total number of cells in the representing bargraph, respectively.For instance, Figure 1 presents the bargraph 122341411.Recently, statistics on bargraphs have been Figure 1: The bargraph 122341411 received a lot of attention.For instance, in [10,17] it is found the generating function for the electronic journal of combinatorics 26(2) (2019), #P2.30 the number of bargraphs according to the number of horizontal and up steps.In [14] it is studied the generating function for the number of bargraphs according to the number of interior vertices and edges.Moreover, Blecher et al. counted bargraphs according to statistics: descents [3], levels [1], peaks [2] and walls [4].
The enumeration of polyominoes according to their area and perimeter is an interesting problem in combinatorics [7,9].When one studies combinatorial families presented geometrically, the perimeter and the area are the most natural and most important statistics to be considered.The perimeter of a bargraph B, denoted by per(B), is the number of edges on the boundary of B. The site-perimeter of a bargraph B, denoted by sper(B), is the number of nearest-neighbor cells outside the boundary of B. The perimeter and the site perimeter of words were studied in [5] and [6], respectively.Motivated by these results, we extend the study of perimeter/site-perimeter to set partitions.
The aim of this paper is to study the perimeter (see Section 2) and the site-perimeter (see Section 3) of set partitions.For example, the perimeter and the site-perimeter of the set partition 122341411 are 32 and 24, respectively.In particular, we show that the total of the half of the perimeter over all set partitions of [n] is given by (see Corollary 5) where B n is the nth Bell number.
In order to obtain asymptotic estimates for the moments as well as the limiting distribution, we need uniformly for h = O(log n), where B n is the nth Bell number and r is the positive root of re r = n + 1. See [8] for an even stronger form that includes further terms in the asymptotic expansion.We say that the sequence f n is asymptotically equivalent to the sequence g n , denoted by f n ∼ g n , if lim n→∞ fn gn = 1.So, which, by ( 1), leads to the following corollary.

The perimeter of set partitions
Let P k (x, q) = n k π∈P n,k x n q 1 2 per(π) be the generating function for the number of set partitions of n with exactly k blocks, according to the half of the perimeter.Generally, let per(π) be the generating function for the number of set partitions π = π a 1 a 2 • • • a s of n with exactly k blocks, according to the half of the perimeter.We define P 0 (x, q) = 1.Since each set partition with one block has the form 11 1−xq .By definitions we have for all 1 a k − 1.Moreover, the electronic journal of combinatorics 26(2) (2019), #P2.30 Define P k (x, q, v) = k a=1 P k (x, q|a)v a−1 .Then ( 2)-( 3) can be written as Thus, we can state the following result.
Proposition 2. We have Proposition 2 with q = 1 gives , as expected.Thus, for all k 1, Define . By Proposition 2, we have . Thus, by using (4) and taking v → 1, we obtain with F 0 (x, 1) = 0. Thus by induction, we can state the following result.
the electronic journal of combinatorics 26(2) (2019), #P2.30 Theorem 3. The generating function F k (x, 1) for the total of the half of the perimeter over all set partitions of [n] with exactly k blocks is given by In order to study further the total of the half of the perimeter over all set partitions of [n], we consider the exponential generating function , where [x n ]f (x) denotes the coefficient of x n in the generating function f (x).Note that n! , where S n,k denotes the Stirling number of the second kind (for example, see [12]).So, by (5), we have with F 0 (x, 1) = 0. Thus, the exponential generating function E k (x) satisfies,

The site-perimeter of set partitions
Let Q k (x, q) = n k π∈P n,k x n q sper(π) be the generating function for the number of set partitions of n with k blocks according to the site-perimeter.Generally, let x n q sper(π) be the generating function for the number of set partition π = π a 1 a 2 • • • a s of n with k blocks according to the site-perimeter.We define Q 0 (x, q) = 1.Since each set partition with one block has the form 11 Since each set partition with two blocks has either the form 11 • • • 12π 2 or the form 11 • • • 12π 1, where π is a word over alphabet {1, 2}, we obtain By the definitions we have where 1 b a − 1 and 1 a k − 1.Moreover, where 1 b k − 1.
Our goal is to find a recurrence relation for the generating function To do that, let H k (x) = x k k j=1 (1−jx) to be the generating function for the number of set partitions of n with exactly k blocks (see [12]).Clearly, the electronic journal of combinatorics 26(2) (2019), #P2.30where 1 b a − 1 and 1 a k − 1.Thus, by ( 10)- (11), we have which, by substituting into Q k (x|a) and using ( 10)- (11), implies for all a = 1, 2, . . ., k − 1.Therefore, by summing over a = 1, 2, . . ., k − 1 we obtain Now, we focus on Q k (x|k).By differentiating ( 8)-( 9), we have where 1 b k − 1.Thus, by substituting expression of Q k (x|bk) into expression of Q k (x|k) with using ( 10)- (11), we obtain By substituting ( 13) into (12), we obtain that the generating function for all k 3. Note that Q 0 . By (14), we can introduce first values of the total of the site-perimeter over all set partitions of [n] with k blocks, see To study further the total of the site-perimeter over all set partitions of [n], we consider the exponential generating function R . By ( 14), we have n! , where S n,k denotes the Stirling number of the second kind (for example, see [12]).Thus, the exponential generating function R k (x) satisfies, R k (x) = k which is equivalent to Remark 8. Based on this work, we realized that there is a typo in the statement of Corollary 2.6 in [13].More precisely, by (1) we have that, asymptotically, the total number of interior vertices (a vertex in B is called an interior vertex if it is adjacent to exactly four different cells of bargraph B) in set partitions of [n + 1]is given by We end the paper by emphasizing that all the above results have been compared with exact enumerations.
y k .By multiplying by y k , and summing over k n , where S n,k denotes the Stirling number of the second kind and B n denotes the nth Bell number.