离散偏移
Discrete excursions
摘要：
众所周知，Dyck路径（步长为+1和-1）的长度母函数E（t）满足1-E+t^2E^2=0。高度不超过k的Dyck路的母函数E^（k）（t）是E^（k）=F_k/ F_{k+1}，其中F_k是t中的多项式，由F_0=F_1=1和F_{k+1} = F_k-t^2 F_{k-1}给出。这意味着这些多项式的母函数是\sum_{k\ge 0} F_k z^k= 1/（1-z+t^2z^2）。我们注意到这个分数的分母是代数级数E（t）的极小多项式。此模式对于具有更常规步数的行走持续存在。对于任何有限的步数集S，在S中取步数且高度至多为k的偏移（广义Dyck路径）的生成函数E^（k）（t）是两个多项式的比F_k/ F_{k+1}。这些多项式满足系数在Q[t]中的线性递归关系。它们的（有理）母函数可以写成\sum_{k\ge 0} F_k z^k= N（t，z）/D（t，z）。偏移生成函数E（t）是代数的并且满足D（t，E（t））=0（当N（t，E（t））\not = 0）。如果max S=a和min S=b，多项式D（t，z）和N（t，z）可以分别取为d_{a,b} =二项式（a+b，a）和d_{a,b} -a-b。这些次数通常是最优的：例如，当S={a，-b}且a和b互质时，D（t，z）是不可约的，因此是偏移生成函数E（t）的极小多项式。这些结果的证明涉及组合和代数工具的略微不寻常的混合，其中包括核方法（解决某些函数方程），对称函数，和一小撮伽罗瓦论。
英文摘要：
It is well-known that the length generating function E(t) of Dyck paths (excursions with steps +1 and -1) satisfies 1-E+t^2E^2=0. The generating function E^(k)(t) of Dyck paths of height at most k is E^(k)=F_k/F_{k+1}, where the F_k are polynomials in t given by F_0=F_1=1 and F_{k+1}= F_k-t^2F_{k-1}. This means that the generating function of these polynomials is \sum_{k\ge 0} F_k z^k= 1/(1-z+t^2z^2). We note that the denominator of this fraction is the minimal polynomial of the algebraic series E(t). This pattern persists for walks with more general steps. For any finite set of steps S, the generating function E^(k)(t) of excursions (generalized Dyck paths) taking their steps in S and of height at most k is the ratio F_k/F_{k+1} of two polynomials. These polynomials satisfy a linear recurrence relation with coefficients in Q[t]. Their (rational) generating function can be written \sum_{k\ge 0} F_k z^k= N(t,z)/D(t,z). The excursion generating function E(t) is algebraic and satisfies D(t,E(t))=0 (while N(t,E(t))\not = 0). If max S=a and min S=b, the polynomials D(t,z) and N(t,z) can be taken to be respectively of degree d_{a,b}=binomial(a+b,a) and d_{a,b}-a-b in z. These degrees are in general optimal: for instance, when S={a,-b} with a and b coprime, D(t,z) is irreducible, and is thus the minimal polynomial of the excursion generating function E(t). The proofs of these results involve a slightly unusual mixture of combinatorial and algebraic tools, among which the kernel method (which solves certain functional equations), symmetric functions, and a pinch of Galois theory.
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