Palindromic and Colored Superdiagonal Compositions

. A superdiagonal composition is one in which the i -th part or sum-mand is of size greater than or equal to i . In this paper, we study the number of palindromic superdiagonal compositions and colored superdiagonal compositions. In particular, we give generating functions and explicit combinatorial formulas involving binomial coeﬃcients and Stirling numbers of the ﬁrst kind.

A palindromic (or self-inverse) composition is one whose sequence of parts is the same whether it is read from left to right or right to left.For example, the palindromic compositions of 4 are (4), (2,2), (1, 2, 1), (1, 1, 1, 1).[8] studied palindromic compositions having parts in a subset of positive integers.In particular, they showed that the total number of palindromic compositions of n is given by 2 n/2 .Moreover, the number of palindromic compositions of n with parts 1 and 2 are the interleaved Fibonacci sequence 1, 1, 2, 1, 3, 2, 5, 3, 8, 5, 13, 8, 21, . . .The literature contains several generalizations and restrictions of the compositions.Much of them are related to the kind of parts or summands, for example compositions with even or odd parts [7,8], with parts in arithmetical progressions [1,9], compositions with colored parts [2,4,11], colored palindromic compositions [3,6,10], superdiagonal compositions [5], among other restrictions.For further information on compositions, we refer the reader to the text by Heubach and Mansour [7].
Deutsch et al. [5] proved that the total number of superdiagonal compositions is given by the combinatorial sum: Agarwal [2] generalized the concept of a composition by allowing the parts to come in various colors.By a colored composition of a positive integer n we mean a composition σ = (σ 1 , σ 2 , . . ., σ ) such that the part of size i can come in one of i different colors.The colors of the summand i are denoted by subscripts i 1 , i 2 , . . ., i i for each i ≥ 1.For example, the colored compositions of 4 are given by (4 1 ), ( 42 ), (4 3 ), (4 4 ), (3 Volumen 56, Número 2, Año 2022 A colored superdiagonal composition is a colored composition such that the i-th part σ i satisfies σ i ≥ i, for each i ≥ 1.The colored superdiagonal compositions of 4 are (4 1 ), ( 42 ), (4 3 ), (4 4 ), ( The goal of this paper is to enumerate the palindromic superdiagonal compositions and colored superdiagonal compositions.These counting sequences are encoded by (ordinary) generating functions.Next, we expand the generating functions as power series to obtain explicit combinatorial formulas for the counting sequences.

Enumeration of the Palindromic Superdiagonal Compositions
Let S Pal denote the set of palindromic superdiagonal compositions.The composition (σ 1 , σ 2 , . . ., σ ) of n can be represented as a bargraph of columns, such that the i-th column contains σ i cells for 1 ≤ i ≤ .For example, in  Let σ be a composition and let us denote the weight of σ by |σ| and the number of parts of σ by ρ(σ).Using these parameters, we introduce this bivariate generating function In Theorem 2.1 we give an expression for the generating function S(x, y).
Theorem 2.1.The bivariate generating function S(x, y) is given by Proof.Let σ = (σ 1 , σ 2 , . . ., σ 2m ) be a palindromic superdiagonal composition of n with 2m parts.From the definition we have the condition σ i = σ 2m+1−i ≥ 2m + 1 − i, for i = 1, . . ., m, see Figure 2 for a graphical representation of this case.The columns i-th and (2m + 1 − i)-th contribute to the generating function the term Therefore the composition σ contributes to the generating function the term If the number of parts is odd, that is σ = (σ 1 , σ 2 , . . ., σ 2m−1 ), then from a similar argument the contribution to the generating function is given by Summing in the above two cases over m we obtain the desired result.

In Theorem 2.2 we give a combinatorial expression for the sequence s(n, m).
We set the convention, n k = 0 whenever k < 0 or k > n.Theorem 2.2.The number of palindromic superdiagonal compositions (1) of 2n with 2k parts equals (2) of either 2n or 2n − 1 with 2k − 1 parts equals Proof.From the proof of Theorem 2.1 we have The combinatorial formula for s(n, 2k − 1) is obtained in a similar manner.

Colored Superdiagonal Compositions
In this section we give the generating function for the total number of colored superdiagonal compositions.Remember that a composition σ = (σ 1 , . . ., σ ) of n is a colored superdiagonal composition if σ i ≥ i for all 1 ≤ i ≤ , with the additional condition that if a part is of size i then it can come in one of i different colors.For example, (3 2 , 2 1 , 5 3 , 5 2 , 6 6 ) is a colored superdiagonal composition of 21.
Let c(n) denote the number of colored superdiagonal compositions of n.In Theorem 3.3 below we give the generating function for this sequence.We need some definitions and one lemma.
Given integers n, k ≥ 0, let n k denote the (unsigned) Stirling numbers of the first kind, which are defined as connection constants in the polynomial identity The Stirling numbers of the first kind give the number of permutations on n elements with k cycles.It is also known that this sequence satisfies the recurrence with the initial conditions 0 0 = 1 and n 0 = 0 n = 0 for n ≥ 1.Let n be a non-negative integer.We introduce the polynomials defined by with the initial value Q 0 (x) = 1.The first six polynomials are Notice that Q n (x) is a polynomial of degree n − 1.Moreover, from (4) we have Lemma 3.1.The polynomials Q n (x) can be expressed as where Proof.From (2), we have the equality Therefore, for all positive integer n we have The first few values of the sequence T (n, k) are This array corresponds with the sequence A059364 in [12].In Theorem 3.2 below we give a combinatorial interpretation for the sequence T (n, k).We will assume that permutations are expressed in standard cycle form, i.e., minimal elements first within each cycle, with cycles arranged left-to-right in ascending order of minimal elements.For example, the permutation 146285937 has the standard cycle form (1)(24)(3658)(79). Notice that we also write the fixed points (cycles of size 1).The symbol i in a permutation is called an overline element.
Theorem 3.2.The sequence T (n, k) counts the number of permutations on n elements such that in the standard cycle form there are k non-minimal overline elements.
Proof.Let π be a permutation on n elements with i cycles (0 ≤ i ≤ n).Notice that there are n i of these permutations.The permutation π has i minimal elements.From the remaining n − i elements, we have to choose the k overline elements in n−i k ways.Summing over i the count is now )(2 3).
Theorem 3.3.The generating function for the number of colored superdiagonal compositions is Proof.Let σ = (σ 1 , . . ., σ m ) be a non-empty colored superdiagonal composition of n with m parts.If σ i = with ≥ i, then σ i contributes to the generating function the term x , for ≥ i and 1 ≤ i ≤ m.Let C m (x) be the generating function of the colored superdiagonal compositions with m parts.Then we have the following expression Finally, summing the last expression over m ≥ 0, we get the desired result.From Theorem 3.3 and Lemma 3.1 we obtain the following corollary.

Revista Colombiana de Matemáticas
Notice that Equation (1) shows the colored superdiagonal compositions corresponding to the bold term in the above sequence.The sequence c(n) does not appear in [12].
Figure 1 we show the superdiagonal compositions of n = 10 with their bargraph representations.
we show the first few values of the sequence s(n, m).