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1h
comment Enumerating matrices function of ranks
@Turbo: The Tao-Vu 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
comment 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
comment How to recognize if a lattice is distributive?
The condition that the posets of join- and meet-irreducibles 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 $n$ join-irreducibles and $n$ meet-irreducibles, then $L$ is distributive.
Jan
15
comment 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
comment 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
comment Is this property of the Bell's number evident?
Another reference is fq.math.ca/Scanned/19-2/gessel.pdf.
Jan
2
answered asymptotic for restricted partitions
Dec
28
awarded  Nice Answer
Dec
28
comment 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
comment Notions of positivity for q-polynomials
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 $\Phi_1(q)=q-1$.
Dec
25
comment 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 $\mathrm{GL}(n-1,q)$ with no entry equal to 1, then $f(n,q)=(q-1)^{2n-1}g(n,q)$. 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
comment 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 log-concave. This is true for any box $(i,j)$, not just a corner box.
Dec
23
comment On a positivity property of Hall-Littlewood polynomials
I found a simple proof that if $\lambda=(\lambda_1,\dots,\lambda_n)$, where $\lambda_i\geq \lambda_{i+1}+n-1$ for all $1\leq i\leq n-1$, 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
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.
Dec
21
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/….