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By Flajolet P., Sedgewick R.

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This condition is equivalent to the existence of a bijection from A to B that preserves size, and one also says that A and B are bijectively equivalent. We normally identify isomorphic classes and accordingly employ a plain equality sign (A = B). We then confine the notation A ∼ = B to stress cases where combinatorial isomorphism results from some nontrivial transformation. 4. The ordinary generating function (OGF) of a sequence (An ) is the formal power series (4) A(z) = ∞ An z n . n=0 The ordinary generating function (OGF) of a combinatorial class A is the generating function of the numbers An = card(An ).

14. Compositions into primes. The additive decomposition of integers into primes is still surrounded with mystery. For instance, it is not known whether every even number is the sum of two primes (Goldbach’s conjecture). 2, p. 297. 5. Partitions with restricted summands (denumerants). Whenever summands are restricted to a finite set, the special partitions that result are called denumerants. A denumerant problem popularized by P´olya [476, §3] consists in finding the number of ways of giving change of 99 cents using coins that are pennies (1 c), nickels (5 c), dimes (10 c) and quarters (25 c).

The proofs are simple verifications from the definitions. ✁ 26 I. 5. Natural numbers. Let Z := {•} with • an atom (of size 1). Then I = S EQ(Z) \ {ǫ} is a way of describing positive integers in unary notation: I = {•, • •, •••, . }. The corresponding OGF is I (z) = z/(1 − z) = z + z 2 + z 3 + · · · . 6. Interval coverings. Let Z := {•} be as before. Then A = Z + (Z × Z) is a set of two elements, • and (•, •), which we choose to draw as {•, •–•}. Then C = S EQ(A) contains objects like •, • •, •–•, • •–•, •–• •, •–• •–•, • • • • .

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