MSET + partitions gen. funct. + Integers gen funct + refactoring and …

…renaming

src/AnalyticComb.jl

@@ -6,14 +6,14 @@ using Reexport
 @reexport using SymPy
 
 export
-	SEQ,
-	stirling_factorial, stirling_catalan,
-	partitions, primes_composition, 
+	SEQ, MSET,
+	stirling_factorial_asym, stirling_catalan_asym,
+	I_gf, partitions_gf, partitions_asym, primes_composition_asym, 
 	restricted_sum_comp_gf, restricted_sum_comp, restricted_sum_part_gf, restricted_sum_part,
 	p_binary_words_doub_runl
 
 include("operators.jl")
-include("asymptotics.jl")
+include("misc.jl")
 include("parts_comps.jl")
 include("binary_words.jl")
 

src/binary_words.jl

@@ -18,8 +18,6 @@ function W_coeff(r;n_tot=200)
 end
 
 
-
-
 """
     p_binary_words_doub_runl(k,n)
 

src/asymptotics.jl → src/misc.jl

@@ -1,18 +1,18 @@
 """
-    stirling_factorial(n)
+    stirling_factorial_asym(n)
 
 Stirling approximation for n! as (n/exp(1))^n*sqrt(2*pi*n).
 
 ``n! \\sim \\sqrt{2 \\pi n} {\\frac{n}{e}}^{n}``
 """
-stirling_factorial(n) = (n/exp(1))^n*sqrt(2*pi*n)
+stirling_factorial_asym(n) = (n/exp(1))^n*sqrt(2*pi*n)
 
 """
-    stirling_catalan(n)
+    stirling_catalan_asym(n)
 
 Stirling approximation for n_th Catalan number .
 
 ``frac{4^n}{\\sqrt{\\pi n^3}}``
 """
-stirling_catalan(n) = 4^n/(sqrt(pi*n^3))
+stirling_catalan_asym(n) = 4^n/(sqrt(pi*n^3))
 

src/operators.jl

@@ -1,6 +1,26 @@
 """
     SEQ(z)
 
-Sequence operator
+Sequence operator (Pólya quasi-inverse operator). 
+
+Defined as ``A = SEQ(B) \\implies A(z) = \\frac{1}{1 - B(z)}``.
+"""
+function SEQ(z)
+    z = SymPy.symbols("z")
+    return(1/(1-z))
+end
+
+
+"""
+    MSET(z)
+
+Multiset operator (Pólya exponential operator). 
+
+Defined as ``A = MSET(B) \\implies A(z) = \\frac{1}{1 - B(z)}``.
 """
-SEQ(z) = 1/(1-z)
+function MSET(z,max)
+    z = SymPy.symbols("z")
+    n = SymPy.symbols("n")
+    return(exp(summation(1/n * z^n,(n,1,max))))
+end
+#MSET(z) = prod(1-z^n)^((-1)*(-B_n))

src/parts_comps.jl

@@ -1,17 +1,42 @@
 """
-    partitions(n)
+    I_gf(z)
+
+Integers as combinatorial structures ``I(z)= \\sum{n \\geq 1}_{} z^n = \\frac{z}{1-z}``
+"""
+function I_gf(z)
+    z = SymPy.symbols("z")
+    return(z*SEQ(z))
+end
+
+"""
+    partitions_gf(z,max)
+
+Generating function for partitions ``P(z)= \\prod{m = 1}_{\\Inf} \\frac{1}{1-z^m}``.
+For instance, use `series(partitions_gf(z,10),z,0,8)` to obtain counts for n up to 8 (EIS A000041)
+"""
+function partitions_gf(z,max)
+    z = SymPy.symbols("z")
+    n = SymPy.symbols("n")
+    prod([1/(1-z^n) for n in 1:max])
+end
+#MSET(I_gf(z)) yields the correct gen. function, as in page 41 (equation 38), 
+# but series(MSET(I_gf(z))) returns coefficients in 2^n
+
+
+"""
+    partitions_asym(n)
 
 Asymptotics for partition of integers by Hardy and Ramanujan, later improved by Rademache
 """
-partitions(n) = (1/(4*n*sqrt(3)))*exp(pi*sqrt(2n/3))
+partitions_asym(n) = (1/(4*n*sqrt(3)))*exp(pi*sqrt(2n/3))
 
 
 """
-    primes_composition(n)
+    primes_composition_asym(n)
 
 Asymptotics for composition of primes.
 """
-primes_composition(n) = 0.30365 * 1.47622^n
+primes_composition_asym(n) = 0.30365 * 1.47622^n
 
 """
     restricted_sum_comp_gf(r)