Richard Stanley
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 11h answered The number of submodules of $\mathbb{Z}_q^n$ 2d comment Is combinatorial automorphism of symmetric convex polytope always odd? I discovered that Fedor's question goes back to Perles. See page 1367 of Kalai's article in Proc. ICM, Zurich, 1994. Nov 21 awarded Yearling Nov 18 answered What are bounds for the number of monotone functions $M:P\rightarrow T$ where $P$ is a finite poset and $T$ is a finite totally ordered set? Nov 16 awarded Good Answer Nov 15 comment Is combinatorial automorphism of symmetric convex polytope always odd? Fedor means that every pair of opposite vertices is sent to a pair of opposite vertices. The paper link.springer.com/article/10.1007%2FBF02760511#page-1 lends credence to a negative answer. Nov 15 revised Is this a rational function? trivial typo corrected Nov 10 comment How to divide a square into three similar rectangles Not relevant to the question, but I cannot resist mentioning the following amazing result of Laczkovich and Szekeres, Discrete Comput. Geom. 13 (1995), 569-572. Let $x>0$. Then a square can be tiled with finitely many copies of rectangles similar to a $1\times x$ rectangle if and only if $x$ is an algebraic number all of whose conjugates have positive real part. For instance, $x=\sqrt{2}+\frac{17}{12}$ is o.k. but not $\sqrt{2}+\frac 43$. Nov 6 comment Simplex in convex polytope, pulling triangulation Pulling triangulations actually go back to Hudson, Piecewise Linear Topology, 1969, Lemma 1.4, and were used by various other researchers before me. Nov 5 comment Linear relations among permutation matrices Oops, you are right. I am thinking of linear relations $A=B$ which involve only positive coefficients. I consider such a relation to be dependent on others if it is a positive linear combination of these other relations. Nov 5 comment Linear relations among permutation matrices For $n=4$ there are 178 independent linear relations. We can also ask for higher order relations (syzygies). More precisely, let $R_n$ be the semigroup algebra (over $\mathbb{R}$, say) of all $n\times n$ matrices of nonnegative integers with equal row and column sums, under addition. Regard $R_n$ as a quotient of the polynomial ring $A_n$ in $n!$ variables, which map to the permutation matrices. Then the Betti numbers of $R_n$ as an $A_n$-module are 1 178 1837 7416 16440 25144 35562 42204 35562 25144 16440 7416 1837 178 1. For $n=5$ the sum of the Betti numbers is $>5.7\times 10^{34}$. Nov 5 comment Volume of the convex hull of the set of all graphic sequences of a given length Actually, I compute the volume of the convex hull of all ordered degree sequences of length $n$, that is, I don't assume $d_1\leq d_2\leq \cdots \leq d_n$. The question of Kozerenko is completely different and seems to be much more difficult. Nov 2 comment Positivity of coefficients of the inverse of a certain power series Here is a conjectured generalization. Let $\Phi_n(x)$ be the $n$th cyclotomic polynomial. Without loss of generality we may assume that $n$ is squarefree. Let $f_n(k)$ be the coefficient of $x^k$ in $\Phi_n(x)^{â(n+1)kâ1}$. If $n$ has an even number of distinct prime divisors, then $f_n(k)>0$. Otherwise $(â1)^kf_n(k)>0$. Nov 1 answered When can one infer degrees of generators of a ring from its hilbert series Oct 31 answered When is the diagonal of a rational bivariate power series again rational Oct 29 awarded rt.representation-theory Oct 29 answered Is there a nice form for the Frobenius characteristic of a border shape character? Oct 22 comment Lexicographic order on increasing $k$-tuples $i$ is any number $0,1,\dots,n-1$. Oct 19 comment Lexicographic order on increasing $k$-tuples @Alessandro: to get your order from mine, read my order from right-to-left: $234, 134, 034, \dots, 012$. Then reverse each word: $432, 431, 430, \dots, 210$. Then replace $i$ with $n-1-i$: $012, 013, 014, \dots, 234$. Oct 19 comment How to prove $\prod_{\lambda\vdash n}\prod_im_i(\lambda)!=\prod_{\lambda\vdash n}1^{m_1(\lambda)}2^{m_2(\lambda)}\cdots$ See Enumerative Combinatorics, vol. 1, 2nd ed., Exercise 1.80. (Incidentally, I don't understand the comment of Ben Barber.)