[Paper Review] Generalized Compositions of Natural Numbers

Milan Janjić|arXiv (Cornell University)|Dec 16, 2010

Advanced Mathematical Theories and Applications3 citations

TL;DR

This paper introduces generalized compositions of natural numbers where each integer i can appear in b_i distinct types, deriving recurrence relations and closed-form formulas for the number of such compositions. It establishes connections to Fibonacci, Catalan, Pell, Jacobsthal, and other classical number sequences, showing that specific choices of b_i sequences yield known integer sequences and enabling explicit formulas via linear recurrences with constant coefficients.

ABSTRACT

We consider compositions of natural numbers when there are different types of each natural number. Several recursions as well as some closed formulas for the number of compositions is derived. We also find its relationships with some known classes of integers such as Fibonacci, Catalan, Pell, Pell-Lucas, and Jacobsthal numbers.

Motivation & Objective

• To generalize standard integer compositions by allowing multiple types (colors) for each part size i, indexed by b_i.
• To derive recurrence relations and closed-form expressions for the number of such generalized compositions.
• To explore connections between generalized compositions and well-known integer sequences such as Fibonacci, Catalan, Pell, and Jacobsthal numbers.
• To identify conditions on the sequence b = (b_1, b_2, ...) that yield linear recurrences with constant coefficients, enabling explicit formulas.
• To demonstrate how known sequences in the OEIS arise as counts of generalized compositions under specific b_i assignments.

Proposed method

• Define c(n, b) as the number of generalized compositions of n, where each part i can be chosen in b_i different types.
• Establish a recurrence: c(n, b) = sum_{i=1}^{n} b_i * c(n-i, b), with c(1, b) = b_1 and c(n, b) = b_n for n > 1 when only one part.
• Introduce a sequence a_n = c(n-1, b), leading to the recurrence a_{n+1} = sum_{i=1}^n b_{n+1-i} a_i.
• Analyze special cases: b_i arithmetic progression, b_i = i^2 (quadratic), b_i exponential, and b_i = Fibonacci numbers.
• Use identities for Fibonacci numbers (e.g., Cassini’s identity, sum identities) to derive closed forms for c(n, b) in these cases.
• Apply Zeilberger’s algorithm to derive identities involving sums of binomial coefficients in the case of triangular b_i.

Experimental results

Research questions

• RQ1 How can standard integer compositions be generalized to allow multiple types (colors) for each part size i, and what recurrence governs the number of such compositions?
• RQ2 Under what conditions on the sequence b_i does the number of generalized compositions satisfy a linear recurrence with constant coefficients?
• RQ3 Which classical number sequences (e.g., Fibonacci, Catalan, Pell, Jacobsthal) emerge as counts of generalized compositions for specific choices of b_i?
• RQ4 Can closed-form expressions be derived for generalized compositions when b_i follows an arithmetic progression, quadratic, or exponential pattern?
• RQ5 What combinatorial identities can be derived by interpreting generalized compositions as weighted sums of binomial coefficients?

Key findings

• When b_i = p for all i, the number of generalized compositions is c(n, b) = p(1+p)^{n-1}, generalizing the standard 2^{n-1} compositions.
• If b_i = C_{i-1} (Catalan numbers), then c(n, b) = C_n, showing that Catalan numbers are fixed points under this composition map.
• For b_i = m(i-1) + k (arithmetic progression), c(n, b) satisfies a three-term linear recurrence with constant coefficients: c(n+1, b) = (k+2)c(n, b) + (m-k-1)c(n-1, b).
• When b_i = i^2 (quadratic), the number of compositions satisfies a four-term linear recurrence with constant coefficients, enabling explicit closed forms.
• For b_i = F_{m + k(i-1)} (Fibonacci-based), the number of compositions satisfies a second-order linear recurrence with coefficients involving Fibonacci numbers.
• Specific choices of m and k in the Fibonacci-based b_i sequence generate known OEIS sequences: e.g., m=0,k=1 gives Jacobsthal numbers (A001045), m=1,k=1 gives Pell numbers (A000129), and m=1,k=2 gives A007052.