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.

本文在发现新的准多项式\（{{\ mathbb {Z}}} \）的数目\（P_K（N）\）的分区的Ñ至多与零件ķ。方法自始至终都是基本的。我们推导出少量多项式（例如，一个用于\(k=3\)，两个用于\(k = 4\)或 5，六个用于\(k=6\)），在添加适当的常数项之后，取值\(p_k(n)\)。例如，对于\(0\le r < 6\)和所有\(q \ge 0\)，\(p_3(6q+r) = p_3(r)+\pi _0(q,r)\)，q和r中总次数为 2 的多项式。一般来说有\(M_{\lfloor k/2 \rfloor } =\) lcm \(\{1,2,\ldots ,\lfloor k/2 \rfloor \}\)这样的多项式。在两个变量q和s 中，它们的形式为\(\sum a_{i,j}{q \atopwithdelims ()i}{s \atopwithdelims ()j}\)和\(a_{i,j} \in {{\mathbb {Z}}}\)，我们称之为整数值多项式的正确形式。它们构成了一个准多项式周期的\（M _ {\ lfloor K / 2 \ rfloor} \）的序列\（（P_K（n）的-p_k（R））\）与\（N \当量r \ PMOD { M_k}\)。对于每个k，所有多项式中最高总次数的项都相同，并且系数仅取决于ķ。第二个定理，结合部分分数和上述方法，找到\({{\mathbb {Q}}}\)上\(p_k(n)\) 的混合多项式比上面的更容易确定。我们将我们的结果与 Cayley、MacMahon 和 Arkin 的结果进行了比较，他们的经典结果在这里重制后非常有效。我们还讨论了 Munagi 的最近结果并得出结论，某种形式的循环器是不可避免的。在\(k=6\) 处，我们发现 Sylvester 计算他的“波浪”时存在严重错误。Sylvester JJ（QJ Pure Appl Math 1:141–152, 1855）。结果被推广到称为“做出改变”的（差别不大）问题，在该问题中发现了对现有方法的重大改进。我们找到了\(p_k(n)\)的无限新同余对于\(k= 3, 4\)和一个新的\(k=5\)。减少模中号的周期序列\（（P_K（N））\）被追究周期性和零：我们发现，从零起步，已知结果在特殊情况下的一个简单证明。