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2019-04-11 08:42:31 -0500 | commented answer | Setting t=0 in a non-symmetric E-Macdonald polynomial Big thanks for this very pedagogical answer! |

2019-04-11 08:33:37 -0500 | answered a question | Constructing all NE-lattice paths from $(0,0)$ to $(m,n)$ I fixed this myself but in case anyone is interested I will leave this as an answer. I decided to use the kbits from this thread to generate all binary strings of length $n$ with $m$ ones. Such strings are in a natural bijection with $NE$-lattice paths from $(0,0)$ to $(n,m)$. After that, I can get the heights of all the paths by writing Note: Sage has a "height" function on words, but this is not what I am looking for. In the definition of height that I am using, the height of an $NE$-path is geometrically the biggest distance that the path is away from the main diagonal. |

2019-04-11 02:56:49 -0500 | asked a question | Setting t=0 in a non-symmetric E-Macdonald polynomial Suppose I have a non-symmetric E-Macdonald polynomial indexed by, say, $\mu=(0,1,1)$. Then I can write and I get a polynomial in three variables and with coefficients in $\mathbb{Q}(q,t)$: However, I am confused about how I can work with this polynomial. For my purposes, I would like to study the specialization $t=0$. It would be really neat if there were some way to get write something like so I could easily specialize variables as I go along. |

2019-04-11 01:50:27 -0500 | asked a question | Constructing all NE-lattice paths from $(0,0)$ to $(m,n)$ If I consider only Dyck Paths, I can do write something like this: to obtain the height of all Dyck paths of length $3$. However, I would like to do the same thing but using $NE$-lattice paths from $(0,0)$ to $(m,n)$. Is there an easy way to do this in Sage? |

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2019-03-23 06:26:52 -0500 | asked a question | Evaluating polynomials in Z[x] at roots of unity. Suppose that I have a polynomial $p(x)$ with integer coeffcients, then I can easily evaluate this at some integer to get an exact answer. That is, I can write something like But If I instead want to evaluate this polynomial at a root of unity (I take $w$ to be a primitive $5$:th root of unity in the example below but any root of unity would do), then I may write say Which is a correct answer but written in a very complicated form. Alternatively, I can write which is an approximation of the correct answers. In the above case, we actually have $p(w)=w^2$ and I am wondering if there is an easy way to always obtain such an answer. For a general $p(x)$ with integer coefficients and $w$ that is a primitive $n$:th root of unity, we can write $p(w)$ on the form $$a_0+a_1 w + a_2 w^2 + \dots +a_{n-1} w^{n-1}$$ where the $a_i$:s are integers. Is there a good way to get the $p(w)$ on this form in Sage? |

2019-03-19 10:36:27 -0500 | commented question | Major index of skew-SYT Added the definition of major index. |

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2019-03-19 07:29:57 -0500 | asked a question | Major index of skew-SYT For a standard Young tableaux (SYT), I can compute the major index as follows: Is there a way to compute the major index of a skew-SYT currently? Something like: At the moment, it seems major_index() is not defined for the SkewTableau-class. EDIT: The major index of a (skew) standard Young tableau $T$, denoted $\mathrm{maj}(T)$, is defined as follows. A descent of $T$ is an entry $i$ such that $i+1$ appears strictly below $i$ in $T$. Define $\mathrm{maj}(T)$ as the sum of all descents of $T$. For example, the major index of the skew standard Young tableau above is $2+4=6$. FindStat has a definition, although they only define it for non-skew tableau. But the definition extends trivially to skew-shapes as well. |

2019-03-12 05:26:15 -0500 | commented answer | Obtaining a poset from Posets(n) This partially solved my issues. But I would, if possible, like to use the major_index-function on the posets of Poset(n). So I would like to write something like `P = Posets(3) Q = next(iter(P)) major_index(Q)` but this gives me a similar error. |

2019-03-10 08:53:21 -0500 | asked a question | Obtaining a poset from Posets(n) I have defined function that takes a poset $P$ as input and outputs the major index generating polynomial over all linear extensions of $P$. Here, the major index of a certain linear extension of $P$ is given by the major index of the corresponding permutation, see e.g. http://mathworld.wolfram.com/LinearEx... for an example of this correspondence. If I write, for example Then I get which is the correct output. I would like to use Posets(n) to generate all posets (up to isomorphism) with $n$ vertices. However, when I write I get the error: Clearly the error is due to the line But I don't understand how Posets(n) works and why I cannot just write lst[3] to obtain a poset. |

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2019-03-05 11:13:27 -0500 | asked a question | Obtaining integers from a linear extension of a poset. I'm quite new to Sage so this might be a very easy question. Suppose that I have defined a linear extension of a poset in Sage like this: Is there a way for me to make this linear extension into a permutation? That is, I would like to obtain the permutation [1,4,2,3] from p. |

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