Search Results

It is well known that the number of rows in the semistandard Young tableaux correspondent to a two-line array via RSK is equal to the length of the longest (strictly) decreasing subsequence in the arr …

This is, probably, a question for those knowledgeable on the subject of Brion's theorem and its applications. Lately, I've been dealing with situations of the following sort. Suppose we are given a p …

Consider a connected skew Young diagram in the English notation and then rotate it counterclockwise by $\pi/4$. This rotation can be avoided by simply replacing "rows" by "diagonals" in the below, but …

One context in which this relationship between the hypercube and the permutahedron appears is as follows. The space of weights of the Lie algebra $\mathfrak{sl}_n$ is naturally identified with $\mathb …

I have a (directed graded) graph $G=(V,E)$ with two distinguished subsets of $V$: sources and destinations. I am to show that the set of indicator functions of paths starting in a source and ending in …

Here's the new, more thought through version. Consider a sequence of nonnegative integers $\lambda=(\lambda_1,\ldots,\lambda_n)$ with $\lambda_i\ge \lambda_{i+1}+2$ (the weight $\lambda-2\rho$ is dom …

Let us consider the matrix $A$ with its rows and columns enumerated by the elements of $S_n$ with $A_{\sigma\tau}=x^{c(\sigma\tau^{-1})}$ where $c()$ is the number of cycles in a permutation's decompo …

Long story short, I personally find Macdonald's celebrated book Symmetric Functions and Hall Polynomials somewhat difficult to read for various reasons. I also know for a fact that I'm not the only on …

For a finite poset $P$ the chain polytope $\mathscr C(P)\subset\mathbb{R}^P$ consists of such $g$ that $g(p)\ge 0$ for all $p\in P$ and $$g(p_1)+\ldots+g(p_n)\le 1$$ for any chain $p_1<\ldots<p_n$. I …

For a root system $\Phi$ of rank $n$ with Weyl group $W$ and a dominant integral weight $\lambda$ consider the Hall-Littlewood polynomial $$P_\lambda(x_1,\ldots,x_n;t)=\frac1{W_\lambda(t)}\sum_{w\in W …

A well known property of finite modular lattices is that they have the same number of join-irreducible and meet-irreducible elements. I was wondering if there exists a natural bijection between these …

I have recently become aware of the following neat statement. Consider a convex polygon $P$ in the real plane with integral vertices. If we associate with every integral point $(a,b)$ the monomial $x …

This is a question I've been meaning to ask for quite some time. Fact. For $n\in\mathbb N$ consider the set of segments $R=\{[i,j], 1\le i<j\le n\}$. Let a subset $E\subset R$ be nice iff $E$ is clos …

I'm interested in connected matroids $M$ on the ground set $[n]$ for which there is no connected matroid on $[n]$ of the same rank but with a strictly smaller set of bases (by inclusion). Equivalently …

Consider a polynomial ring $\mathbb C[x_1,\ldots,x_n]$ that is $\mathbb Z_{\ge 0}$-graded by degree. Let $I$ and $J$ be two homogeneous ideals therein with the same Hilbert series, i.e. with their hom …