Richard Stanley

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 Name Richard Stanley Member for 3 years Seen 1 hour ago Website Location Age
 1d answered Why are the dinv-statistic and the partition length equidistributed? May17 revised how to proof this Stirling related equationcorrected spelling of Stirling May16 awarded ● Stellar Question May9 comment Discrete disjoint covering of integer latticesFor any $n\times n$ integer matrix $M$ with determinant $d\neq 0$ and columns $v_1,\dots,v_n$, there are exactly $|d|-1$ nonzero integer vectors $u_1,\dots,u_{d-1}$ of the form $u_i=\sum a_i v_i$, where $0\leq a_i<1$. The point here is that these nonzero $u_i$'s should form a basis for the lattice $\mathbb{Z}^n$. May9 answered Discrete disjoint covering of integer lattices May7 accepted Semi-Standard Young Diagrams and Families May7 answered Semi-Standard Young Diagrams and Families Apr28 awarded ● Nice Answer Apr25 comment Inequivalence of group representations preserved under tensor product?If $r_1$ and $r_2$ are two ordinary linear representations of the same degree of a finite group $G$ and $R$ is the regular representation, then $r_1\otimes R\cong r_2\otimes R$. Apr24 comment Using extended group rings for combinatorial generating functionsGuoce Xin has some papers that seem relevant. See for instance front.math.ucdavis.edu/0409.5190 and front.math.ucdavis.edu/0504.5425. Apr10 comment Criteria for ghost-Witt vectors: looking for history and referencesSome partial information (with references) can be found in Exercise 5.2 of Enumerative Combinatorics, vol. 2. Apr9 accepted Relations involving Stirling numbers of second kind Apr9 answered Relations involving Stirling numbers of second kind Apr5 comment Access to a preprint by D. N. VermaThe above formula can be simplified to $$\frac 12\sum_{j=0}^{m-1}(m-j)^{n-3}(-1)^{j+1}\binom nj.$$ Mar27 comment Other Variant of Schur Polynomials/FunctionsIgnoring your list of properties, there is a list of some variants of Schur functions (with references) in the Notes to Chapter 7 of my book Enumerative Combinatorics, vol. 2. Mar21 comment Counting seating arrangements at a circular tableDo the members of a team sit together with no spaces between them? In your example, since the boys are identical it seems to me that there is only one way of seating two teams of two and one way of seating a team of three and a team of one, so two ways in all. Mar21 answered Counting linear extensions of unlabeled series parallel structures Mar19 comment Characters of p-groupsIn fact, if $M$ is the number of nonlinear irreducible characters, then $M\equiv 0$ (mod $p^2-1$) if $n\equiv r$ (mod 2); $M\equiv p-1$ (mod $p^2-1$) if $n$ is odd and $r$ is even; and $M\equiv -p+1$ (mod $p^2-1$) if $n$ is even and $r$ is odd. Mar19 accepted Characters of p-groups Mar18 answered Characters of p-groups Mar16 answered Generalization’s of Greene’s Theorem for the Robinson-Schensted correspondence Mar8 comment Invertibility of a certain matrix indexed by the Hamming cubeThe solution to Exercise 3.96(a) shows how the matrix $M=F(s\wedge t,s)$ can be triangularized, from which a recurrence for the entries of $M$ follows. I believe this is essentially what is done in Section 6 of arXiv:1110.4954, though I haven't looked at this very carefully. It would be interesting to find an explicit formula for the entries of $M^{-1}$. Mar8 answered Invertibility of a certain matrix indexed by the Hamming cube Mar2 comment Access to a preprint by D. N. VermaI would like to thank Yannic Vargas, who located a copy of Verma's paper in the LACIM library and emailed me a scanned file. Mar1 awarded ● Nice Question Feb28 comment Access to a preprint by D. N. VermaThanks, this is very helpful. For $n=2m$ I get the formula $$\frac 12\sum_{j=0}^{m-1}(m-j)^{n-3}(-1)^{j+1}{n\choose j} (2j-n+1).$$ Assuming that my sequence and Verma's are the same (which must be true), then this gives a simpler formula than in the reference you provide. Feb28 asked Access to a preprint by D. N. Verma Feb21 revised Good books on problem solving / math olympiadbroken link fixed Feb18 accepted Statistics on Lehmer codes Feb18 answered Statistics on Lehmer codes Feb16 answered What is the probability for sequence of lenght L in subset of [n] Feb15 accepted Partial order relation on subsets Feb14 answered Partial order relation on subsets Feb9 comment quotients of polynomial ringsIf the ideal $I$ is generated by homogeneous polynomials, then the condition is that $I$ is generated by polynomials of degree one. Feb9 revised quotients of polynomial ringsmissing ] inserted Feb6 awarded ● Nice Answer Jan25 comment Is there a characterization of hyperplane arrangement intersection posets?Just the question of when a geometric lattice is the intersection lattice of a linear hyperplane arrangement is quite complicated. It is equivalent to asking when a matroid can be represented over a field. Jan22 answered connected components of a real hyperplane arrangement Jan19 awarded ● Good Question Jan15 answered Elementary applications of linear algebra over finite fields Jan13 awarded ● Popular Question Jan8 awarded ● Nice Question Jan5 comment A question on the Laurent phenomenon@quid: not at all. Thanks for doing this. Jan5 asked A question on the Laurent phenomenon Jan4 comment power of adjacency matrixIt is highly unlikely that there is any simple computation to decide whether there is a path of length $\ell$ between two vertices (much less count how many such paths there are), since the existence of a path of length $p-1$, where $G$ has $p$ vertices, is NP-complete. Dec30 comment Graph of $S_n$ with respect to transpositionThe graph $G_n$ is the Hasse diagram of the absolute order on $S_n$. This gives some insight into what this graph "looks like." See arxiv.org/pdf/0706.1405v2.pdf. Dec22 comment Reconstructing the argument that yields Graham’s numberThough irrelevant to Tim's question, Graham's number is small potatoes compared to some of the numbers cooked up by Harvey Friedman, e.g., his paper Long finite sequences, JCT(A) 95 (2001), 102-144. Dec22 comment Any non-conforming numbers?Every integer is of the form $x^2-y^2$ or $x^2-y^2+1$, and thus of the form $x^2-y^2+z^p$ for any $p$ (where we can take $z=0$ or $z=1$), so you need to rule out this situation. Dec21 comment Reference: Finite $p$-GroupsThe book Enumeration of Finite Groups by Blackburn, Neumann, and Venkataraman has a lot of information on $p$-groups. Dec21 accepted Are there any binomial poset which has non-isomorphic interval of the same length?
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