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15h
comment Maximum connected components $0-1$ matrix
I didn't realize from the way you stated the problem above that you were allowing row and column permutations. You already raised your question in mathoverflow.net/questions/190981.
17h
comment Maximum connected components $0-1$ matrix
You can get $k=\lceil n/2\rceil^2$ by taking $a_{ij}=1$ whenever $i$ and $j$ are both odd, and this is the maximum.
23h
comment On a positivity property of Hall-Littlewood polynomials
@Igor: I haven't seen this statement before. Some random checking suggests that an even stronger statement is true: under your condition on $\lambda$, the expansion of $P_\lambda(x;-t)$ in terms of Schur functions has coefficients that are polynomials in $t$ with nonnegative coefficients.
1d
comment On a positivity property of Hall-Littlewood polynomials
One has to distinguish between the $P$-Macdonald basis (which specializes to Hall-Littlewood by setting $q=0$) and the $H$-Macdonald basis (for which Haglund, Haiman, and Loehr gave a combinatorial interpretation of the expansion into monomials). See sagemath.org/doc/reference/combinat/sage/combinat/sf/….
1d
comment Connected components $0-1$ matrices
Why is this question on hold? It seems quite interesting to me.
Dec
13
awarded  Nice Question
Dec
13
comment On a positivity property of Hall-Littlewood polynomials
We have $P_{2,2}(x;-t)=m_{2,2}+(t+1)m_{2,1,1}+(-t^3+3t+2)m_{1,1,1,1}$. Doesn't this contradict your assertion? On the other hand, it is well-known that $P_\lambda(x;-1)$ is Schur-positive.
Dec
13
comment Lots of combinatorial interpretations of Catalan numbers
Thanks for your answer. I realize that my question is not very precise, but your answer seems like a reasonable solution.
Dec
12
asked Lots of combinatorial interpretations of Catalan numbers
Dec
12
comment Marked chain polytope, has this been studied?
There is no simple hook-like formula for the volume. See Theorem 1 and Theorem 11 of the paper cited in my answer.
Dec
11
answered Marked chain polytope, has this been studied?
Dec
10
comment A sum-of-determinants identity
Doesn't this follow easily from the multilinearity of the determinant?
Dec
9
comment Non-representable irreducible matroid of rank at least 5?
Any irreducible matroid of rank 5 or higher that contains the Vamos matroid (en.wikipedia.org/wiki/Vámos_matroid) as a minor will do.
Nov
21
awarded  Yearling
Nov
19
answered Is there any relationship between the topologies of the clique complex and the independence complex?
Nov
16
comment Number of orders of $k$-sums of $n$-numbers
@MaxAlekseyev: Actually, the fourth term of A231085 is 2. This is because A231085 has the additional condition $x_1<x_2<\cdots<x_n$. To get from the $n$th term of A231085 to the sequence I am considering, multiply by $n!$.
Nov
12
awarded  Nice Answer
Nov
8
comment Number of orders of $k$-sums of $n$-numbers
An arrangement in $\mathbb{R}^n$ with $m$ hyperplanes has at most $1+m+{m\choose 2}+\cdots+{m\choose n}$ regions. This gives a bound not far from what you want. More careful reasoning (for instance, using the fact that the arrangement is central) should give a better bound. For background on hyperplane arrangements, see math.mit.edu/~rstan/arrangements/arr.html.
Nov
8
answered Number of orders of $k$-sums of $n$-numbers
Nov
2
comment Who first noticed that Stirling numbers of the second kind count partitions?
@David. Interesting. Thanks for checking this.