# bell_polynomial(n,k).coefficients() raises an error

Consider the Bell polynomials:

for n in (0..4):
print [bell_polynomial(n,k) for k in (0..n)]

[1]
[0, x_1]
[0, x_2, x_1^2]
[0, x_3, 3*x_1*x_2, x_1^3]
[0, x_4, 3*x_2^2 + 4*x_1*x_3, 6*x_1^2*x_2, x_1^4]


Extracting the coefficients I expect this triangle:

[[1]]
[[0], [1]]
[[0], [1], [1]]
[[0], [1], [3], [1]]
[[0], [1], [3, 4], [6], [1]]


However this call (which looks very natural) does not work:

for n in (0..4):
print [bell_polynomial(n,k).coefficients() for k in (0..n)]


It gives the error message: AttributeError: 'sage.rings.rational.Rational' object has no attribute 'coefficients'. However I nowhere give a sage.rings.rational.Rational object as an input. My input is a polynomial if I can trust the name "bell_polynomial(n,k)". So may I consider this as an internal error/bug?

Ralf Stephan showed me a workaround:

def polynomial_coefficients(p):
if (isinstance(p, (int, Integer, Rational))):
return [p]
else:
return p.coefficients()


This gives indeed the expected output. However it might be even better to represent the null polynomial by the empty list (to comply with the understanding that only nonzero coefficients are shown).

if (isinstance(p, (int, Integer, Rational))):
if p == 0 return [] else return [p]


[[1]]
[[], [1]]
[[], [1], [1]]
[[], [1], [3], [1]]
[[], [1], [3, 4], [6], [1]]


Is this a workaround just to fit my possibly ill-guided expectations or is this a reasonable patch for the general case? Or is it correct that an object which is called a polynomial raises an error if asked for the coefficients?

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My guess is that this is yet another example of not knowing what category a number belongs in. 1, 0, etc. are probably given as rationals or integers and not polynomials, because probably whoever implemented this just put them into that. Perhaps in that code for Bell polynomials all output should be explicitly polynomials (which polynomial ring, I don't know)?

( 2015-04-29 20:52:37 +0100 )edit

For what it's worth I describe a different implementation of the partial Bell polynomials (which might be more efficient) here: https://oeis.org/wiki/User:Peter_Lusc...

( 2015-05-05 21:15:00 +0100 )edit

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The cause of your problem is that bell_polynomial(0,0) does not return 1 as a polynomial, but as a rational number.

sage: bell_polynomial(0,0)
1
sage: bell_polynomial(0,0).parent()
Rational Field


I consider this as a bug : bell_polynomial(n,k) should always be an element of a polynomial ring, the doc claims explicitely that the output should be "a polynomial in n-k+1 variables over \QQ". Thanks for reporting. This is now trac ticket 18338 (needs review).

With the patch applied, you get directly:

sage: bell_polynomial(0,0)
1
sage: bell_polynomial(0,0).parent()
Univariate Polynomial Ring in x_1 over Rational Field

sage: for n in (0..4):
....:     print [bell_polynomial(n,k) for k in (0..n)]
[1]
[0, x_1]
[0, x_2, x_1^2]
[0, x_3, 3*x_1*x_2, x_1^3]
[0, x_4, 3*x_2^2 + 4*x_1*x_3, 6*x_1^2*x_2, x_1^4]

sage: sage: for n in (0..4):
....:     print [bell_polynomial(n,k).coefficients() for k in (0..n)]
[[1]]
[[], [1]]
[[], [1], [1]]
[[], [1], [3], [1]]
[[], [1], [3, 4], [6], [1]]

more

All polynomials with k=0 are affected, and the error is not even consistent: b = bell_polynomial(3,0); b.parent() leads to: AttributeError: 'int' object has no attribute 'parent'. tmontail, is there a convention how the coefficients of the null polynomial are represented in Sage? [] or 0?

( 2015-04-30 07:51:40 +0100 )edit
1

As mentioned in the doc, the coefficients() method for polynomials provides the list of .nonzero coefficients, hence the expected result should be [].

( 2015-04-30 12:09:39 +0100 )edit
1

The error is consistent in the sense that in the current implementation, it starts with the Python int 0, then it is (sometimes) added with the Sage rational 1, then it is (sometimes) added with some genuine polynomial. The implementation of trac ticket 18338 solves this, you will get:

sage: bell_polynomial(3,0).parent()
Multivariate Polynomial Ring in x_1, x_2, x_3, x_4 over Rational Field

( 2015-04-30 12:11:48 +0100 )edit
1

By the way, as a user of Bell polynomials, would you mind if the indices of the variables start at 0, e.g. if bell_polynomial(3,2) = 3*x_0*x_1 instead of 3*x_1*x_2 ? This woudl be more consistent with other polynomials in Sage.

( 2015-04-30 13:38:57 +0100 )edit

Thank you tmonteil! I agree absolutely with this. More in my answer at the track ticket.

( 2015-04-30 19:40:56 +0100 )edit