Packed words and quotient rings

The coinvariant algebra is a quotient of the polynomial ring $\mathbb{Q}[x_1,\dots ,x_n]$ whose algebraic properties are governed by the combinatorics of permutations of length n. A word $w=w_1\dots w_n$ over the positive integers is packed if whenever $i>2$ appears as a letter of w, so does $i?1$. We introduce a quotient $S_n$ of $\mathbb{Q}[x_1,\dots ,x_n]$ which is governed by the combinatorics of packed words. We relate our quotient $S_n$ to the generalized coinvariant rings of Haglund, Rhoades, and Shimozono as well as the superspace coinvariant ring.     

Bargraph statistics on words and set partitions

In this paper, we consider statistics on partitions of an n-element set represented as a subset of the bargraphs that have n horizontal steps. More precisely, we find the joint distribution of the area and up step statistics on the latter subset of bargraphs, thereby obtaining new refined counts on partitions having a fixed number of blocks. Furthermore, we give explicit formulas in terms of the Stirling numbers for the total area and number of up steps in bargraphs corresponding to partitions, providing both algebraic and combinatorial proofs. Finally, we find asymptotic estimates for the average and total values of these statistics and as a consequence obtain some new identities for the Stirling and Bell numbers.     

Identification and definition of lexically ambiguous words in statistics by tutors and students

Lexical ambiguity arises when a word from everyday English is used differently in a particular discipline, such as statistics. This paper reports on a project that begins by identifying tutors’ perceptions of words that are potentially lexically ambiguous to students, in two different ways. Students’ definitions of nine lexically ambiguous words are also collected at the beginning and end of a semester of introductory statistics study, in a complex design taking account of multiple tutors and multiple words in multiple contexts. Tutor perceptions and actual student difficulties at the beginning of a semester are compared. The lexical ambiguity associated with the word ‘significance’ is shown to be evident in students even after completing an introductory statistics course.     

The regularity of munn rings and semigroup rings

In this paper, the author studies the regularity of Munn rings and completely 0-simple semigroup rings. When either (i)R is a ring with identity andI and are infinite, or (ii)R is a strong IBN and fully Dedekind-finite ring with identity and eitherI or is finite, in Section 2, the regularity of the Munn ringM(R; I, ; P) is characterized; in Section 3, for a completely 0-simple semigroupS =M°(G; I, ; P), the regularity ofRS is characterized. Meantime, the author shows that for a locally finite monoidS and a ringR with identity,RS is a strong IBN ring if and only ifR is so;RS is fully Dedekind-finite if and only ifR is so.     

The perimeter of words

We define $\left[k\right]=\{1,2,\dots ,k\}$ to be a (totally ordered) alphabet on $k$ letters. A word $w$ of length $n$ on the alphabet $\left[k\right]$ is an element of ${\left[k\right]}^n$. A word can be represented by a bargraph (i.e., by a column-convex polyomino whose lower edges lie on the $x$-axis) in which the height of the $i$th column equals the size of the $i$th part of the word. Thus these bargraphs have heights which are less than or equal to $k$. We consider the perimeter, which is the number of edges on the boundary of the bargraph. By way of Cramer’s method and the kernel method, we obtain the generating function that counts the perimeter of words. Using these generating functions we find the average perimeter of words of length $n$ over the alphabet $\left[k\right]$. We also show how the mean and variance can be obtained using a direct counting method.     

Relations on words

In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation.In the second part, we mainly focus on abelian equivalence, $k$-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and $M$-equivalence. In particular, some new refinements of abelian equivalence are introduced.     

环的余挠维数

研究了Milnor方图上的余挠维数,然后探讨了环的余挠维数和整体维数,弱整体维数之间的关系和差别.证明了一个Prüfer整环的余挠维数不超过1当且仅当它是整体维数不超过2的Matlis整环.     

A note on the coefficient rings of polynomial rings

LetA andB be two reduced commutative rings with finitely many minimal prime ideals. If the polynomial algebrasA[X ₁ …X _n ]=B[Y ₁ …Y _n ] whereX _i ,Y _iF are variables overA andB respectively, then there exists an injective ring homomorphism ϕ:A→B such thatB is finitely generated over ϕ(A).     

The holomorphy rings of planar ternary rings with rational prime field

By a deep, recent result of Hartmann and Priess-Crampe any ordered planar ternary ring with rational prime field admits a natural order-compatible place into the reals. This allows us to extend the classical machinery about real places and spaces of orderings, the notion of real and relative holomorphy rings, the Kadison-Dubois representation for these rings, and the Brown-Marshall inequality to arbitrary planar ternary rings with rational prime field.     

The generic division rings

LetA=k (X ₁, X₂..., X_m) be the division ring generated by genericn×n matrices over a fieldk; thenA is not a crossed product in the following cases: (i) there exists a primeq such thatq ³ℛn;(ii)[k:Q]=m, whereQ is the field of rationals, then if eitherq ³ℛn for someq for whichq-1ℛm, orq ²/nn for some other prime; (iii)k=Z _p ^r a finite field ofp ^r elements and eitherq ³ℛn for sameqℛp ^r-1 orq ²ℛn for some other primes. Other cases are also considered.     

The cancellable range of rings

We unify the cancellation property of rings with stable range one and the principal ideal domain by introducing a new notion which is called “cancellable range”. It is proved that if a ring R has cancellable range n for some positive integer n, then for any n-generated module B and any module implies B ≅ C; if R is a Noetherian ring and R has cancellable range n for any n ≧ 1, then R has the cancellation property. Received: 16 November 2004     

The cancellation property of rings

A ring R is said to have the finitely generated cancellation property provided that the module isomorphism R⊕B?R⊕C implies B?C for any finitely generated R-modules B and C. It is proved that R has this property is equivalent to the existence of the cancellation matrices over R. Moreover, the structure of such matrices is investigated and finite weakly stable rings are characterized in terms of their cancellation matrices.     

Rank functions on rings derived from group rings

The ring D_G derived from the group ring KG is shown to posses exactly one pseudo-rank function in case G is an infinite, but locally and residually finite, group and K is a field. The function is described naturally as the limit of the sequence of normalized ranks of all approximation matrices. This fact is applied to an investigation of the direct finiteness of the maximal quotient ring of D_G .     

The Auslander–Reiten translate on monomial rings

For t in Nn, E. Miller has defined a category of positively t-determined modules over the polynomial ring S in n variables. We consider the Auslander–Reiten translate, Nt, on the (derived) category of such modules. A monomial ideal I is t-determined if each generator xa has a?t. We compute the multigraded cohomology and Betti spaces of for every iterate k and also the S-module structure of these cohomology modules. This comprehensively generalizes results of Hochster and Gräbe on local cohomology of Stanley–Reisner rings.