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1h

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Enumerating matrices function of ranks
@Turbo: The TaoVu paper only gives improved bounds. It is still quite far from an asymptotic determination. See Conjecture 1.3, which is still open. Perhaps there is an analogue of Conjecture 1.3 for rank $r$. See also arxiv.org/pdf/math/0501313.pdf. 
3h

accepted  A congruence involving binomial coefficients 
23h

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Enumerating matrices function of ranks
The probability that a random $n\times n$ $(0,1)$matrix is singular is equal to the probability that a random $(n+1)\times (n+1)$ $(1,1)$matrix is singular (by a simple argument). For the status of this difficult problem, see arxiv.org/pdf/math/0411095v3.pdf. 
1d

awarded  Nice Question 
1d

asked  A congruence involving binomial coefficients 
Jan 26 
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How to recognize if a lattice is distributive?
The condition that the posets of join and meetirreducibles are isomorphic is certainly not sufficient for distributivity, e.g., a $k$element antichain, $k>2$, with a top and bottom adjoined. I don't know whether the following is true: let $L$ be a finite lattice with maximal chain of length $n$. If $L$ has exactly $n$ joinirreducibles and $n$ meetirreducibles, then $L$ is distributive. 
Jan 15 
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How to recognize if a lattice is distributive?
I am not sure this question deserves to be on hold. The question of how to recognize which lattices are distributive from their diagram is rather interesting. In addition to the classical result about $M_3$ and $N_5$, there is the result of Farley and Schmidt that a finite lattice is distributive if and only if every open interval is either an antichain or is connected, and every interval of rank three is distributive. (Up to isomorphism, there are five distributive lattices of rank three.) See Enumerative Combinatorics, vol. 1, 2nd ed., Exercise 3.31. 
Jan 15 
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How to recognize if a lattice is distributive?
Birkhoff's theorem states that a modular lattice generated by two chains is distributive. It is not true that any lattice generated by two chains is distributive. 
Jan 5 
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Is this property of the Bell's number evident?
Another reference is fq.math.ca/Scanned/192/gessel.pdf. 
Jan 2 
answered  asymptotic for restricted partitions 
Dec 28 
awarded  Nice Answer 
Dec 28 
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Trying to prove a congruence for Stirling numbers of the second kind
There is a similar result for Stirling numbers of the first kind. See Corollary 3.4 of math.mit.edu/~rstan/papers/cycles.pdf. 
Dec 26 
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Notions of positivity for qpolynomials
If you want the set of all "positive" polynomials to form a cone, then the smallest such cone is genererated by products of cyclotomic polynomials $\Phi_n(q)$ excluding $\Phi_1(q)=q1$. 
Dec 25 
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Number of elements of “$\mathrm{SL}_n(\mathbb{F}_p^\times)$” mod $p$
Not relevant to your question, but Exercise 1.182 of Enumerative Combinatorics, vol. 1, 2nd ed., shows that if $f(n,q)$ is the number of matrices in $\mathrm{GL}(n,q)$ with no 0 entries, and $g(n,q)$ is the number of matrices in $\mathrm{GL}(n1,q)$ with no entry equal to 1, then $f(n,q)=(q1)^{2n1}g(n,q)$. A reference for counting matrices in $\mathrm{GL}(n,q)$ with specified entries equal to 0 (though it doesn't seem useful for your question) is arXiv:1011.4539. 
Dec 25 
revised 
Functions representable as a sum of two permutations of Z/nZ
clarification added 
Dec 25 
answered  Functions representable as a sum of two permutations of Z/nZ 
Dec 23 
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Number of standard Young tableaux with fixed corner entry
It is an immediate consequence of the case $k=1$ of Theorem 3.1 of math.mit.edu/~rstan/pubs/pubfiles/48.pdf that the sequence $N_{ij}(1), N_{ij}(2),\dots,N_{ij}(n)$ is logconcave. This is true for any box $(i,j)$, not just a corner box. 
Dec 23 
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On a positivity property of HallLittlewood polynomials
I found a simple proof that if $\lambda=(\lambda_1,\dots,\lambda_n)$, where $\lambda_i\geq \lambda_{i+1}+n1$ for all $1\leq i\leq n1$, then the coefficients of the Schur function expansion of $P_\lambda(x_1,\dots,x_n;t)$ are polynomials in $t$ with nonnegative coefficients (which can be described combinatorially). I don't see any way of extending the proof to answer Igor's question. 
Dec 21 
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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. 
Dec 21 
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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/…. 