 By NOAMD.ELKIES

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Extra info for Combinatorial game Theory in Chess Endgames

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Then, a triangulation decomposes into its root triangle and two subtriangulations (that may well be “empty”) appearing on the left and right sides of the root triangle; the decomposition is illustrated by the following diagram: = + 34 I. UNLABELLED STRUCTURES AND ORDINARY GENERATING FUNCTIONS The class T of all triangulations can be specified recursively as T = {ǫ} + (T × ∇ × T ) , provided that we consider a 2-gon (a diameter) as giving rise to an empty triangulation. Consequently, the OGF satisfies the equation T = 1 + zT 2 and √ 1 1 − 1 − 4z .

Then, the OGF of I is, as we know, z , (25) I(z) = zn = 1−z n≥1 since In = 1 for n ≥ 1, corresponding to the fact that there is exactly one object in I for each size n ≥ 1. If integers are represented in unary, say by small balls, one has, (26) I = {1, 2, 3, . } = {•, • •, • • •, . } ∼ = S EQ≥1 {•}, which is another way to view the equality I(z) = z/(1 − z). Compositions. 1, a direct translation into OGF: 1 . (27) C = S EQ(I) =⇒ C(z) = 1 − I(z) The collection of equations (25), (27) thus fully determines C(z): C(z) = 1 1−z = z 1 − 1−z 1 − 2z = 1 + z + 2z 2 + 4z 3 + 8z 4 + 16z 5 + 32z 6 + · · · .

Compositions and partitions. Our first examples have to do with decomposing integers into sums. 9. A composition of an integer n is a sequence (x1 , x2 , . . , xk ) of integers (for some k) such that n = x1 + x2 + · · · + xk , xj ≥ 1. A partition of an integer n is a sequence (x1 , x2 , . . , xk ) of integers (for some k) such that n = x1 + x2 + · · · + xk and x1 ≥ x2 ≥ · · · ≥ xk . In both cases, the xi ’s are called the summands or the parts and the quantity n is called the size of the composition or the partition.