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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?

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)}

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)}

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)}

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)}