AnalyticComb
Documentation for AnalyticComb.
AnalyticComb.I_gfAnalyticComb.MSETAnalyticComb.SEQAnalyticComb.W_coeffAnalyticComb.p_binary_words_doub_runlAnalyticComb.partitions_asymAnalyticComb.partitions_gfAnalyticComb.primes_composition_asymAnalyticComb.restricted_sum_compAnalyticComb.restricted_sum_comp_gfAnalyticComb.restricted_sum_partAnalyticComb.restricted_sum_part_gfAnalyticComb.stirling_catalan_asymAnalyticComb.stirling_factorial_asym
AnalyticComb.I_gf — MethodI_gf(z)Integers as combinatorial structures
$I(z)= \sum_{n \geq 1} z^n = \frac{z}{1-z}$
AnalyticComb.MSET — MethodMSET(z)Multiset operator (Pólya exponential operator).
Defined as $A = MSET(B) \implies A(z) = \frac{1}{1 - B(z)}$.
AnalyticComb.SEQ — MethodSEQ(z)Sequence operator (Pólya quasi-inverse operator).
Defined as $A = SEQ(B) \implies A(z) = \frac{1}{1 - B(z)}$.
AnalyticComb.W_coeff — MethodW_coeff(r;n_tot=200)Taylor series coefficient from generating function for binary words that never have more than r consecutive identical letters.
The number of binary words that never have more than r consecutive identical letters is found to be (set α = β = r). n_tot defaults to 200, according to the example in Flajolet & Sedgewick pag. 52
AnalyticComb.p_binary_words_doub_runl — Methodp_binary_words_doub_runl(k,n)Returns probablity associatied with k-lenght double runs (or either 0s or 1s) in a sequence of size n.
Specification is W ∼= SEQ(b) SEQ(a SEQ(a) b SEQ(b)) SEQ(a). Refer to the example in Flajolet & Sedgewick pag. 52.
AnalyticComb.partitions_asym — Methodpartitions_asym(n)Asymptotics for partition of integers (EIS A000041) by Hardy and Ramanujan, later improved by Rademache
AnalyticComb.partitions_gf — Methodpartitions_gf(z,max)Generating function for integer partitions.
$P(z)= \prod{m = 1}_{\Inf} \frac{1}{1-z^m}$ Use series to obtain counts(EIS A000041): series(partitions_gf(z,10),z,0,8) for n up to 8.
AnalyticComb.primes_composition_asym — Methodprimes_composition_asym(n)Asymptotics for composition of n into prime parts (A023360).
AnalyticComb.restricted_sum_comp — Methodrestricted_sum_comp(n,r)Number of compositions of n with components in the set {1,2,..,r}.
r = 2 yields Fibonnaci numbers (EIS A000045): $F_{n} = F_{n-1} + F_{n-2}$. r>2 yields generalized Fibonacci numbers.
AnalyticComb.restricted_sum_comp_gf — Methodrestricted_sum_comp_gf(r)Generating function for compositions with restricted summand.
AnalyticComb.restricted_sum_part — Methodrestricted_sum_part(n,r)Number of partitions with components in r with restricted summand n.
n must be an integer and r must be a set of integers, like in r = [1,5,10,25] , n = 99.
AnalyticComb.restricted_sum_part_gf — Methodrestricted_sum_part_gf(r)Generating function for partition with restricted summand.
AnalyticComb.stirling_catalan_asym — Methodstirling_catalan_asym(n)Stirling approximation for n_th Catalan number. (EIS A000108)
$\frac{4^n}{\sqrt{\pi n^3}}$
AnalyticComb.stirling_factorial_asym — Methodstirling_factorial_asym(n)Stirling approximation for n! as (n/exp(1))^nsqrt(2pi*n). (EIS A000142)
$n! \sim \sqrt{2 \pi n} {\frac{n}{e}}^{n}$