[Paper Review] Some Formulas for Numbers of Restricted Words

Milan Janjić|arXiv (Cornell University)|Feb 4, 2017

semigroups and automata theory1 references3 citations

TL;DR

This paper introduces a generalized composition function $ c_m(n,k) $ that extends integer compositions into $ k $ parts using iterated inverse transforms of an initial arithmetic function $ f_0 $. It establishes connections between $ c_m(n,k) $, partial Bell polynomials, Pascal matrices, and restricted words over finite alphabets. The key contribution is a recursive transfer method that lifts word-counting properties from an alphabet of size $ N $ to $ N+1 $, enabling enumeration of restricted words via $ c_m(n,k) $, with explicit formulas derived for specific $ f_0 $, such as $ f_0(1)=1 $ or $ f_0(n)=1 $ for $ n>1 $.

ABSTRACT

We define a quantity $c_m(n,k)$ as a generalization of the notion of the composition of the positive integer $n$ into $k$ parts. We proceed to derive some known properties of this quantity. In particular, we relate two partial Bell polynomials, in which the sequence of the variables of one polynomial is the invert transform of the sequence of the variables of the other. We connect the quantities $c_m(n,k)$ and $c_{m-1}(n,k)$ via Pascal matrices. We then relate $c_m(n,k)$ with the numbers of some restricted words over a finite alphabet. We develop a method which transfers some properties of restricted words over an alphabet of $N$ letters to the restricted words over an alphabet of $N+1$ letters. Several examples illustrate our findings. Note that all our results depend solely on the initial arithmetic function $f_0$.

Motivation & Objective

• Develop a generalized composition function $ c_m(n,k) $ that extends standard integer compositions using iterated inverse transforms of an initial arithmetic function $ f_0 $.
• Establish a connection between $ c_m(n,k) $ and partial Bell polynomials, showing that $ c_m(n,k) $ arises from weighted compositions with variables derived from $ f_{m-1} $.
• Derive a matrix-based recurrence using lower triangular Pascal matrices to relate $ c_m(n,k) $ and $ c_{m-1}(n,k) $, enabling recursive computation.
• Provide a general method to transfer properties of restricted words over an alphabet of size $ N $ to those over $ N+1 $ letters, based on the structure of $ f_0 $.
• Offer explicit combinatorial interpretations and closed-form formulas for $ c_m(n,k) $ under specific $ f_0 $, such as $ f_0(1)=1 $ or $ f_0(n)=1 $ for $ n>1 $, linking them to binary and ternary word counts.

Proposed method

• The function $ c_m(n,k) $ is defined recursively via $ c_m(n,k) = \sum_{i=1}^{n-k+1} f_{m-1}(i) c_m(n-i,k-1) $, with base cases $ c_m(0,0)=1 $, $ c_m(n,0)=0 $ for $ n>0 $, generalizing standard compositions.
• Prove that $ c_m(n,k) $ equals $ \frac{k!}{n!} B_{n,k}(1!f_{m-1}(1), 2!f_{m-1}(2), \ldots) $, linking it to partial Bell polynomials and weighted integer compositions.
• Use the invert transform relationship between sequences to relate $ c_m(n,k) $ and $ c_{m-1}(n,k) $, showing that the transformation corresponds to multiplication by a lower triangular Pascal matrix.
• Derive a transfer formula via Equation (9), expressing $ c_m(n,k) $ as a sum over $ t $, where $ c_m(n,k) = \sum_{t=0}^{n-k} {k+t-1 \choose t} c_{m-1}(n,k+t) $, enabling recursive lifting from $ m-1 $ to $ m $.
• Apply the method to specific $ f_0 $: when $ f_0(1)=1 $, $ c_1(n,k) $ counts compositions into parts $ \geq 2 $, and $ c_m(n,k) $ counts restricted words over $ \{0,1,\ldots,m\} $ with isolated nonzero letters.
• Use combinatorial bijections and binomial identities to derive closed-form expressions, such as $ c_m(n,k) = \sum_{j=0}^{\lfloor n/2 \rfloor - k} (m-1)^j {j+k-1 \choose k-1} {n-k-j-1 \choose k+j-1} $, under $ f_0(n)=1 $ for $ n>1 $.

Experimental results

Research questions

• RQ1 How can the standard notion of integer composition into $ k $ parts be generalized to incorporate weighted or restricted compositions via iterated inverse transforms?
• RQ2 What is the relationship between $ c_m(n,k) $ and partial Bell polynomials, and how does this connection reveal the combinatorial structure of weighted compositions?
• RQ3 How can the properties of restricted words over an alphabet of size $ N $ be systematically extended to alphabets of size $ N+1 $ using the function $ c_m(n,k) $?
• RQ4 What explicit formulas can be derived for $ c_m(n,k) $ when the initial function $ f_0 $ is specified, such as $ f_0(1)=1 $ or $ f_0(n)=1 $ for $ n>1 $?
• RQ5 How do Pascal matrices and binomial identities facilitate the recursive computation and structural analysis of $ c_m(n,k) $ for varying $ m $ and $ k $?

Key findings

• The function $ c_m(n,k) $ satisfies $ c_m(n,n) = f_{m-1}(1)^n $, showing that diagonal entries depend only on the first value of the $ (m-1) $-th inverse transform.
• Equation (4) establishes that $ c_m(n,k) = \frac{k!}{n!} B_{n,k}(1!f_{m-1}(1), 2!f_{m-1}(2), \ldots) $, proving that $ c_m(n,k) $ is a normalized partial Bell polynomial in the variables $ i!f_{m-1}(i) $.
• Equation (5) shows that $ c_m(n,k) = \sum_{i_1+\cdots+i_k=n} f_{m-1}(i_1)\cdots f_{m-1}(i_k) $, meaning $ c_m(n,k) $ counts weighted compositions of $ n $ into $ k $ positive parts with weights $ f_{m-1}(i) $.
• For $ f_0(1)=1 $, the paper derives $ c_m(n,k) = \sum_{j=0}^{\lfloor n/2 \rfloor - k} (m-1)^j {j+k-1 \choose k-1} {n-k-j-1 \choose k+j-1} $, providing a closed-form for restricted words with isolated nonzero letters.
• Corollary 29 states that $ c_m(n,k) $ counts words of length $ n-3 $ over $ \{0,1,\ldots,m\} $ with $ k-1 $ ones, all isolated, and no two consecutive nonzero letters, when $ f_0(n)=1 $ for $ n>1 $.
• Corollary 31 gives $ c_m(n,k) = \sum_{j=0}^{\lfloor n/2 \rfloor} (m-1)^j {j+k-1 \choose k-1} {k+j \choose n-2k-2j} $ for $ f_0(2)=f_0(3)=1 $, $ f_0(n)=0 $ otherwise, counting words with isolated nonzero letters and no run of zeros longer than 2.