On restricted partitions of numbers

Abstract

This paper finds new quasi-polynomials over ${{\mathbb {Z}}}$ for the number $p_k(n)$ of partitions of n with parts at most k. Methods throughout are elementary. We derive a small number of polynomials (e.g., one for $k=3$, two for $k = 4$ or 5, six for $k=6$) that, after addition of appropriate constant terms, take the value $p_k(n)$. For example, for $0\le r < 6$ and for all $q \ge 0$, $p_3(6q+r) = p_3(r)+\pi _0(q,r)$, a polynomial of total degree 2 in q and r. In general there are $M_{\lfloor k/2 \rfloor } =$ lcm$\{1,2,\ldots ,\lfloor k/2 \rfloor \}$ such polynomials. In two variables q and s, they take the form $\sum a_{i,j}{q \atopwithdelims ()i}{s \atopwithdelims ()j}$ with $a_{i,j} \in {{\mathbb {Z}}}$, which we call the proper form for an integer-valued polynomial. They constitute a quasi-polynomial of period $M_{\lfloor k/2 \rfloor }$ for the sequence $(p_k(n)-p_k(r))$ with $n \equiv r \pmod {M_k}$. For each k the terms of highest total degree are the same in all the polynomials and have coefficients dependent only on k. A second theorem, combining partial fractions and the above approach, finds hybrid polynomials over ${{\mathbb {Q}}}$ for $p_k(n)$ that are easier to determine than those above. We compare our results to those of Cayley, MacMahon, and Arkin, whose classical results, as recast here, stand up well. We also discuss recent results of Munagi and conclude that circulators in some form are inevitable. At $k=6$ we find serious errors in Sylvester’s calculation of his “waves.” Sylvester JJ (Q J Pure Appl Math 1:141–152, 1855). The results are generalized to the (not very different) problem called “making change,” where significant improvements to existing approaches are found. We find an infinitude of new congruences for $p_k(n)$ for $k= 3, 4$, and one new one for $k=5$. Reduced modulo m the periodic sequence $(p_k(n))$ is investigated for periodicity and zeros: we find, from scratch, a simple proof of a known result in a special case.

Appendix

1.1 A. Partial fractions

Here is a brief classical account of partial fractions.

Theorem 4

Let K be a field and, for $m \in {{{\mathbb {Z}}}}_{>0}$, let $f_1(x),\ldots ,f_m(x) \in K[x]$ be relatively prime in pairs. Let $N(x) \in K[x]$ be relatively prime to, and have degree less than that of, $f_1(x)\cdots f_m(x)$.

Then there are uniquely determined polynomials $a_1(x), \ldots , a_m(x) \in K[x]$ such that

$$\begin{aligned} \frac{N(x)}{f_1(x)\cdots f_m(x)} = \sum _{1\le j \le m}\frac{a_j(x)}{f_j(x)}, \end{aligned}$$

(67)

and for each j the degree of $a_j(x)$ is less than that of $f_j(x)$.

From now on we’ll denote the degree of a polynomial f(x) by $\partial f(x)$.

Proof

We sketch the proof of the theorem: use induction on m. There is nothing to prove for $m = 1$. Let $m=2$. There are $a(x), b(x) \in K[x]$ such that

$$\begin{aligned} a(x)f_1(x) + b(x)f_2(x) = N(x). \end{aligned}$$

(68)

In (68) replace a(x) [b(x)] by its remainder r(x) [$r'(x)$] on division by $f_2(x)$ [$f_1(x)$]. The result (if we drop the “(x)” for a moment) is

$$\begin{aligned} rf_1 + r'f_2 -N \equiv 0 \pmod {f_1f_2}; \end{aligned}$$

(69)

a degree argument shows the congruence is equality, and shows the uniqueness. Divide this equality (69) by $f_1f_2$ to get the desired result.

The proof for $m=2$ is in essence the proof of the inductive step. $\square$

Corollary 10

For $e_1,\ldots ,e_m \in {{{\mathbb {Z}}}}_{>0}$, let $D(x):= f_1(x)^{e_1}\cdots f_m(x)^{e_m}$. Let $N(x) \in K[x]$ be prime to D(x) and suppose $\partial N(x) < \partial D(x)$. Then for $j = 1,\ldots , m$ there are uniquely determined $b_j(x) \in K[x]$ such that $\partial b_j(x) < e_j\partial f_j(x)$ and

$$\begin{aligned} \frac{N(x)}{D(x)} = \sum _{1\le j \le m}\frac{b_j(x)}{f_j(x)^{e_j}}, \end{aligned}$$

(70)

Moreover, each numerator $b_j(x)$ may be expanded “in the base $f_j(x)$” to yield

$$\begin{aligned} \frac{N(x)}{D(x)} = \sum _{1\le j \le m}\sum _{1 \le i\le e_j}\frac{c_{ji}(x)}{f_j(x)^i}, \end{aligned}$$

(71)

in which the uniquely determined $c_{ji}(x) \in K[x]$ satisfy $\partial c_{ji}(x) < \partial f_j(x)$.

Math packages for partial fractions typically return equation (71).

Finally, we specialize the Corollary to the case when, for distinct nonzero $\rho _1,\ldots , \rho _m \in K, f_1(x) = 1 - \rho _1x, \ldots , f_m(x) = 1 - \rho _mx$. This case may arise in the study of generating functions, in which one seeks a formula for the coefficient of $x^n$ in the power-series expansion of

$$\begin{aligned} G(x) := N(x)/D(x) \end{aligned}$$

(72)

(notation: [$x^n$]G(x)). We’ll see that [$x^n$]G(x) is a sum of m polynomials in n each multiplied by one of the $\rho$s to the power n.

The $c_{ji}$ in (71) are now constants in K. We use the binomial theorem to write the general term in (71) as

$$\begin{aligned} \frac{c_{ji}}{(1-\rho _j x)^i}=c_{ji}\sum _{n\ge 0}{n+i-1\atopwithdelims ()i-1}{\rho _j}^nx^n. \end{aligned}$$

(73)

Thus

$$\begin{aligned} {[}x^n]\frac{c_{ji}}{(1-\rho _j x)^i} = c_{ji}{n+i-1 \atopwithdelims ()i-1}{\rho _j}^n. \end{aligned}$$

(74)

Summed over i this becomes

$$\begin{aligned} \left\{ c_{j1}+c_{j2}n + \cdots + c_{j,e_j}{n+e_j-1 \atopwithdelims ()e_j-1}\right\} {\rho _j} \end{aligned}$$

(75)

In braces is a polynomial in n of degree $e_j-1$; call it $h_j(n)$. Thus

$$\begin{aligned} {[}x^n]G(x) = \sum _{1\le j\le m}h_j(n){\rho _j}^n; \end{aligned}$$

(76)

moreover, the leading coefficient of $h_j(n)$, the coefficient of $n^{e_j-1}$, is

$$\begin{aligned} \frac{c_{j,e_j}}{(e_j - 1)!} = \frac{N(1/\rho _j)}{(e_j-1)!}\prod _{i\ne j}(1-\rho _i/\rho _j)^{-e_i}, \end{aligned}$$

(77)

which follows from (72) on multiplication by $(1 - \rho _jx)^{e_j}$ and the substitution $x := 1/\rho _j$.

Three other treatments ([6, pp. 279-280], [14, p. 340], [24, pp. 1-2]) of partial fractions differ from the classical treatment, which still stands up well, though that in [24] is interesting.

1.2 B Integer-valued polynomials

Definition

A polynomial over ${{\mathbb {Q}}}$ in n variables $x_1,\ldots ,x_n$ is an integer-valued polynomial iff it maps ${{{\mathbb {Z}}}_{\ge 0}}^n$ to ${{{\mathbb {Z}}}}$.

Proposition 13

Every integer-valued polynomial $W(x_1,\ldots ,x_n)$ is uniquely a sum of terms

$$\begin{aligned} a_\mathbf{k} {x_1 \atopwithdelims ()k_1}{x_2 \atopwithdelims ()k_2}\cdots {x_n \atopwithdelims ()k_n} \end{aligned}$$

for various $\mathbf{k} := (k_1,\ldots ,k_n) \in {{{\mathbb {Z}}}_{\ge 0}}^n$ in which $a_\mathbf{k}$ is an integer.

Proof

Let $m \ge 0$. The space of polynomials in y of degree at most m over ${{\mathbb {Q}}}$ has the two bases $B_1 := \{1,y,\ldots ,y^m\}$ and

$$\begin{aligned} B_2 := \{1, {y \atopwithdelims ()1}, {y \atopwithdelims ()2}, \ldots , {y \atopwithdelims ()m}\}. \end{aligned}$$

Thus every element in $B_1$ is a linear combination of elements of $B_2$ with coefficients from ${{\mathbb {Q}}}$.

Let $\mathbf{x}$ stand for $(x_1,\ldots ,x_n)$, and for $\mathbf{k}$ as above, let

$$\begin{aligned} {\mathbf{x} \atopwithdelims ()\mathbf{k}} := {x_1 \atopwithdelims ()k_1}\cdots {x_n \atopwithdelims ()k_n}. \end{aligned}$$

Our polynomial is now expressible as

$$\begin{aligned} W(\mathbf{x}) = \sum _\mathbf{k}a_\mathbf{k}{\mathbf{x} \atopwithdelims ()\mathbf{k}}, \end{aligned}$$

(78)

the sum taken over various $\mathbf{k} \in {{{\mathbb {Z}}}_{\ge 0}}^n$. The coefficients are in ${{\mathbb {Q}}}$. We now show that those coefficients are integers.

For each $\mathbf{k}$ define its height to be $k_1 + \cdots + k_n$. Note that if $\mathbf{k_1}$ and $\mathbf{k_2}$ have the same height but are not equal, then for each one there is a coordinate in which it is greater than the other.

We start with a $\mathbf{k}= (k_1,\ldots ,k_n)$ of least height in (78). Set $\mathbf{x} = \mathbf{k}$ in (78). The result: $W(\mathbf{k})=a_\mathbf{k}$, since all other terms are 0 (each involves a factor ${u \atopwithdelims ()v}$ in which $0 \le u < v$). We proceed thus through all the coefficients, always choosing next a $\mathbf{k}$ of least remaining height. Integrity is forced on us since each of the as is determined by the prior results and a value of W. Uniqueness is forced by the facts about bases of vector spaces: the polynomial W, given in the variables $x_1,\ldots ,x_n$, determines uniquely the coefficients in (78). $\square$

Corollary 11

Every integer-valued polynomial in n variables maps ${{\mathbb {Z}}}^n$ to ${{\mathbb {Z}}}$.

In this paper we call a formula for an integer-valued polynomial proper if it is in the form of Eq. (78).

Deeper study of these polynomials appears in [10].