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15h

comment 
Maximum connected components $01$ 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

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Maximum connected components $01$ 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

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On a positivity property of HallLittlewood 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

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On a positivity property of HallLittlewood polynomials
One has to distinguish between the $P$Macdonald basis (which specializes to HallLittlewood 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

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Connected components $01$ 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 HallLittlewood 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 wellknown that $P_\lambda(x;1)$ is Schurpositive. 
Dec 13 
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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 
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Marked chain polytope, has this been studied?
There is no simple hooklike 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 
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A sumofdeterminants identity
Doesn't this follow easily from the multilinearity of the determinant? 
Dec 9 
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Nonrepresentable 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 
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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 
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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 
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Who first noticed that Stirling numbers of the second kind count partitions?
@David. Interesting. Thanks for checking this. 