![]() | 1 | initial version |
Let me answer Q#2:
K = 5 # or any other number
A = PolynomialRing(QQ,K,'a')
a = A.gens()
P.<x> = PolynomialRing(A)
f = sum( ai * x^i / factorial(i) for i,ai in enumerate(a) )
print(f)
Is this what you want?
![]() | 2 | No.2 Revision |
Let me answer Q#2:UPDATED. Here a function that computes missing partitions in RT polynomials (based on the code given in MO question):
K = 5 # or any other number
def RT_missing(K):
A = PolynomialRing(QQ,K,'a')
a = A.gens()
#P.<x> = PolynomialRing(A)
P.<x> = PolynomialRing(A)
PowerSeriesRing(A)
f = 1 / (1 + sum( ai * x^i x^(i+1) / factorial(i) factorial(i+1) for i,ai in enumerate(a) )
print(f)
)).add_bigoh(K+1)
g = integrate(f)
h = g.reverse()
RT = (1 / derivative(h,x))[K] * factorial(K)
return { tuple(p.to_exp(K)) for p in Partitions(K) } - { tuple(k) for k in RT.dict().keys() }
Is this what you want?For $K\leq 30$ it produces non-empty sets only for $K\in \{ 7,8,13,14,23 \}$ - namely:
7 {(0, 1, 0, 0, 1, 0, 0)}
8 {(3, 1, 1, 0, 0, 0, 0, 0)}
13 {(0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0)}
14 {(0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0)}
23 {(0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0)}
![]() | 3 | No.3 Revision |
UPDATED. Here a function that computes missing partitions in RT polynomials (based on the code given in MO question):
def RT_missing(K):
A = PolynomialRing(QQ,K,'a')
a = A.gens()
#P.<x> = PolynomialRing(A)
P.<x> = PowerSeriesRing(A)
f = 1 / (1 + sum( ai * x^(i+1) / factorial(i+1) for i,ai in enumerate(a) )).add_bigoh(K+1)
g = integrate(f)
h = g.reverse()
RT = (1 / derivative(h,x))[K] * factorial(K)
return { tuple(p.to_exp(K)) for p in Partitions(K) } - { tuple(k) for k in RT.dict().keys() }
For $K\leq 30$ it produces non-empty sets only for $K\in \{ 7,8,13,14,23 \}$ - namely:
7 {(0, 1, 0, 0, 1, 0, 0)}
8 {(3, 1, 1, 0, 0, 0, 0, 0)}
13 {(0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0)}
14 {(0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0)}
23 {(0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0)}
![]() | 4 | No.4 Revision |
UPDATED. Here a function that computes missing partitions in RT polynomials (based on the code given in MO question):
def RT_missing(K):
A = PolynomialRing(QQ,K,'a')
a = A.gens()
P.<x> = PowerSeriesRing(A)
f = 1 / (1 + sum( ai * x^(i+1) / factorial(i+1) for i,ai in enumerate(a) )).add_bigoh(K+1)
g = integrate(f)
h = g.reverse()
RT = (1 / derivative(h,x))[K] * factorial(K)
return { tuple(p.to_exp(K)) for p in Partitions(K) } - { tuple(k) for k in RT.dict().keys() RT.dict() }
For $K\leq 30$ it produces non-empty sets only for $K\in \{ 7,8,13,14,23 \}$ - namely:
7 {(0, 1, 0, 0, 1, 0, 0)}
8 {(3, 1, 1, 0, 0, 0, 0, 0)}
13 {(0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0)}
14 {(0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0)}
23 {(0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0)}
![]() | 5 | No.5 Revision |
UPDATED. Here a function that computes missing partitions in RT polynomials (based on the code given in MO question):
def RT_missing(K):
RT_poly(K):
A = PolynomialRing(QQ,K,'a')
a = A.gens()
P.<x> = PowerSeriesRing(A)
f = 1 / (1 + sum( ai * x^(i+1) / factorial(i+1) for i,ai in enumerate(a) )).add_bigoh(K+1)
g = integrate(f)
h = g.reverse()
RT =
return (1 / derivative(h,x))[K] * factorial(K)
def RT_missing(K):
return { tuple(p.to_exp(K)) for p in Partitions(K) } - { tuple(k) for k in RT.dict() }
set( RT_poly(K).exponents(False) )
For $K\leq 30$ it 30$, RT_missing(K)
produces non-empty sets only for $K\in \{ 7,8,13,14,23 \}$ - namely:
7 {(0, 1, 0, 0, 1, 0, 0)}
8 {(3, 1, 1, 0, 0, 0, 0, 0)}
13 {(0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0)}
14 {(0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0)}
23 {(0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0)}