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1d
comment All relations among degree n monomials in n variables
You are asking for the first syzygy module of the $n$th Veronese subalgebra of $k[x_1,\dots,x_n]$. A relevant paper is arxiv.org/pdf/1403.4796.pdf.
1d
awarded  Enlightened
1d
awarded  Nice Answer
1d
revised Expected size of determinant of $AA^T$ for non-square random $A$
new paragraph added
1d
answered Expected size of determinant of $AA^T$ for non-square random $A$
Jun
30
comment Counting matrices of special types
@MaxAlekseyev I meant that you apply $(1,2)$ to both the rows and the columns. I see that you meant something else.
Jun
30
comment Counting matrices of special types
@MaxAlekseyev I don't think that your assertion about $k^{i\cdot j}$ is true. Let $k=2$, $\pi=\sigma=(1,2)$. Then the number of matrices invariant with respect to $\pi$ and $\sigma$ is four, not two.
Jun
29
comment Probability that random nonnegative integer matrix is singular
It has been conjectured that the probability that an $n\times n$ $\pm 1$ matrix $M$ is singular is asymptotic to the probability that two rows or columns of $M$ are equal up to sign (which is easy to compute). See users.uoa.gr/~apgiannop/matrices/Kahn_Komlos_Szemeredi_1995.pdf. Perhaps an analogue holds for matrices with entries $0,1,\dots,k$, e.g., two rows or columns being linearly dependent.
Jun
27
comment Is there a nice formula for the “non-crossing substitution” of linear combinatorial species?
Exercise 5.35 of Enumerative Combinatorics, vol. 2, is relevant.
Jun
25
awarded  Good Answer
Jun
25
comment Extrapolation between longest increasing and longest alternating subsequences
@TheMaskedAvenger: I have added the fact that $w=a_1\cdots a_n$. The case $f(n)>n$ is equivalent to $f(n)=n$.
Jun
25
revised Extrapolation between longest increasing and longest alternating subsequences
added "$w=$", "of $w$".
Jun
25
asked Extrapolation between longest increasing and longest alternating subsequences
Jun
24
accepted Combinatorial interpretation of ${i\choose n}$, where $i^2=-1$
Jun
16
awarded  Generalist
Jun
15
comment Finding the square root of a special matrix
If the entries of $A$ are $0,\pm x_i$ and $A$ is a symmetric matrix, then you are asking for a symmetric orthogonal design matrix. One reference is amazon.com/exec/obidos/ASIN/0824767748.
Jun
15
revised Evaluation of Macdonald Polynomials at $x_1=x_2=…=x_n = h$
"Macdonald" misspelled
Jun
12
comment Sum of irreducible complex character degrees for alternating groups
Following up on my previous comment, an interesting related result is that $\sum_\lambda f^\lambda (-1)^{\frac 12(n-\mathrm{rank}(\lambda))}$ equals the coefficient of $x^n/n!$ in $e^{x-\frac{x^2}{2}}$. Here $\lambda$ ranges over all self-conjugate partitions of $n$, $f^\lambda$ is the dimension of the corresponding symmetric group character, and $\mathrm{rank}(\lambda)$ is size of the Durfee square of $\lambda$ (the largest $i$ for which $\lambda_i\geq i$).
Jun
11
answered Representation numbers of numerical semigroups
Jun
10
awarded  Nice Answer