exp-generating-series.at

exp_multiply

exp_mult

the same, but start in degrees |k,l|, and return only relevan part

exp_divide

exp_diff

exp_substitute

f->f[X:=g]

exp_comp_inverse

sin

we can now define some trigonometric functions here; exp is just |ones|

cos

tan

count_permutations_with_cycles

for instance to count permutations with allowed cycles of distinct lengths
given in a list |L|, one can compute $\exp(\sum_{l\in L}{X^l\over l})$,
which by the property of the exponential can be done as follows

also in this file at line 156

exp_of_exponential_series

Here is more general exponentiation

count_permutations_with_cycles

as an example of using |exp_of_exponential_series|, consider the question of
counting permutations with a condition on the cycle length, but which is not
given by a finite list of allowed cycles (for example, any odd length cycles
are allowed). This would require an infinite product to be made in
|count_permutations_with_cycles|, and indeed infinite products can be dealt
with. But the product comes from in infinite sum under the |exp| operator, so
instead we can just produce an exponential generating series and then apply
|exp_of_exponential_series| to it. We do this here, where the argument is an
|inf_list|, supposed strictly increasing and of positive integers, that
describes the set of allowed cycle lengths.

also in this file at line 114

derangement_numbers

for example here is how to count derangements without the alternating formula