SEQ operator + stirling aprox. for Catalan numbers + partitions and c…

…ompositions + partitions with restricted summand + test unit using problem from page 43

Project.toml

@@ -4,6 +4,7 @@ authors = ["fargolo <[email protected]> and contributors"]
 version = "1.0.0-DEV"
 
 [deps]
+Reexport = "189a3867-3050-52da-a836-e630ba90ab69"
 SymPy = "24249f21-da20-56a4-8eb1-6a02cf4ae2e6"
 
 [compat]

README.md

@@ -33,6 +33,8 @@ pkg>add https://github.com/fargolo/AnalyticComb.jl.git
 
 # Quick start  
 
+SymPy.jl functionalities are reexported.  
+
 Probability for consecutive runs of lenght k in binary words of length n, use `p_binary_words_runl(k,n)`
 
 ```
@@ -47,3 +49,14 @@ julia> stirling_factorial(7)
 4980.395831612462
 ```
 
+Using SEQ operator to solve Polya denumerant problem (see Flajolet & Sedgewick,p. 43) about the number of ways of giving change of 99 cents using pennies (1 cent), nickels (5 cents), dimes (10 cents) and quarters (25 cents).  
+```
+julia> restricted_sum_part_gf([1,5,10,25]) # generating function SEQ(z)*SEQ(z^5)*SEQ(z^10)*SEQ(z^25)
+                 1                  
+────────────────────────────────────
+        ⎛     5⎞ ⎛     10⎞ ⎛     25⎞
+(1 - z)⋅⎝1 - z ⎠⋅⎝1 - z  ⎠⋅⎝1 - z  ⎠
+
+julia>restricted_sum_part(99,[1,5,10,25]) 
+213
+```

dev/chap1.jl

@@ -6,6 +6,3 @@ permutation_count(n) = factorial(n)
 # p 20 and 35
 catalan_gen(z) = (1−sqrt(1−4z))/2z
 series(catalan_gen(z),z,0,10)
-
-#p38
-stirling_catalan(n) = 4^n/(sqrt(pi*n^3))

dev/operators.jl

@@ -38,11 +38,13 @@ primes_composition(n) = 0.30365 * 1.47622^n
 
 #change
 using SymPy
-change_counts(z) = 1/((1-z)(1 - z^5)(1 - z^10)(1 - z^25))
-series(change_counts(z),z,0,99)
-coefs = collect(series(change_counts(z),z,0,99),z)
-coefs.coeff(z,99)
-coefs.coeff(z^99)
-coefs.coeffs()
 @vars z
-p.coeffs()
+r = [1,5,10,25]
+restricted_sum_part(r)=prod([1/(1-z^i) for i in r])
+restricted_sum_part(r)
+change_counts(z) = 1/((1-z)*(1 - z^5)*(1 - z^10)*(1 - z^25))
+SEQ(z) = 1/(1-z)
+change_counts(z) = SEQ(z)*SEQ(z^5)*SEQ(z^10)*SEQ(z^25)
+coefs = collect(series(change_counts(z),z,0,100),z)
+coefs = collect(series(restricted_sum_part(r),z,0,100),z)
+coefs.coeff(z,99)

src/AnalyticComb.jl

@@ -1,14 +1,20 @@
 module AnalyticComb
 
-using SymPy
-#using TaylorSeries
-#using DynamicalSystems
+#using SymPy
+using Reexport
+
+@reexport using SymPy
 
 export
-	stirling_factorial, partitions, primes_composition,
+	SEQ,
+	stirling_factorial, stirling_catalan,
+	partitions, primes_composition, restricted_sum_part_gf, restricted_sum_part,
 	p_binary_words_runl
 
-include("binary_words.jl")
+include("operators.jl")
 include("asymptotics.jl")
+include("parts_comps.jl")
+include("binary_words.jl")
+
 
 end

src/asymptotics.jl

@@ -5,18 +5,10 @@ Stirling approximation for n! as (n/exp(1))^n*sqrt(2*pi*n).
 """
 stirling_factorial(n) = (n/exp(1))^n*sqrt(2*pi*n)
 
-
 """
-    partitions(n)
+    stirling_catalan(n)
 
-Asymptotics for partition of integers by Hardy and Ramanujan, later improved by Rademache
+Stirling approximation for n_th Catalan number .
 """
-partitions(n) = (1/(4*n*sqrt(3)))*exp(pi*sqrt(2n/3))
-
+stirling_catalan(n) = 4^n/(sqrt(pi*n^3))
 
-"""
-    primes_composition(n)
-
-Asymptotics for composition of primes.
-"""
-primes_composition(n) = 0.30365 * 1.47622^n

src/binary_words.jl

@@ -36,5 +36,3 @@ function p_binary_words_runl(k,n)
     return(Float64(result_sym))
 end
 
-
-

src/operators.jl

@@ -0,0 +1,6 @@
+"""
+    SEQ(z)
+
+Sequence operator
+"""
+SEQ(z) = 1/(1-z)

src/parts_comps.jl

@@ -0,0 +1,37 @@
+"""
+    partitions(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))
+
+
+"""
+    primes_composition(n)
+
+Asymptotics for composition of primes.
+"""
+primes_composition(n) = 0.30365 * 1.47622^n
+
+"""
+    restricted_sum_part_gf(r)
+
+Generating function for partition with restricted summand.
+"""
+function restricted_sum_part_gf(r)
+    z = SymPy.symbols("z")
+    return(prod([SEQ(z^i) for i in r]))
+end
+
+"""
+    restricted_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.
+"""
+function restricted_sum_part(n,r)
+    z = SymPy.symbols("z")
+    coefs = collect(series(restricted_sum_part_gf(r),z,0,n+1),z)
+    return(coefs.coeff(z,n))
+end

test/runtests.jl

@@ -17,5 +17,6 @@ using Test
     0.0440234813776148,
     0.022607888203734293]) .< 0.01) != 0
 
+    @test restricted_sum_part(99,[1,5,10,25]) == 213 # Polya problem for partition with restricted sum
 
 end