CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt146",{id:"formSmash:upper:j_idt146",widgetVar:"widget_formSmash_upper_j_idt146",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt147_j_idt149",{id:"formSmash:upper:j_idt147:j_idt149",widgetVar:"widget_formSmash_upper_j_idt147_j_idt149",target:"formSmash:upper:j_idt147:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

On the Shape of a Pure O-sequencePrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true}); PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt223",{id:"formSmash:j_idt223",widgetVar:"widget_formSmash_j_idt223",onLabel:"Hide others and affiliations",offLabel:"Show others and affiliations"});
function selectAll()
{
var panelSome = $(PrimeFaces.escapeClientId("formSmash:some"));
var panelAll = $(PrimeFaces.escapeClientId("formSmash:all"));
panelAll.toggle();
toggleList(panelSome.get(0).childNodes, panelAll);
toggleList(panelAll.get(0).childNodes, panelAll);
}
/*Toggling the list of authorPanel nodes according to the toggling of the closeable second panel */
function toggleList(childList, panel)
{
var panelWasOpen = (panel.get(0).style.display == 'none');
// console.log('panel was open ' + panelWasOpen);
for (var c = 0; c < childList.length; c++) {
if (childList[c].classList.contains('authorPanel')) {
clickNode(panelWasOpen, childList[c]);
}
}
}
/*nodes have styleClass ui-corner-top if they are expanded and ui-corner-all if they are collapsed */
function clickNode(collapse, child)
{
if (collapse && child.classList.contains('ui-corner-top')) {
// console.log('collapse');
child.click();
}
if (!collapse && child.classList.contains('ui-corner-all')) {
// console.log('expand');
child.click();
}
}
PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2012 (English)In: Memoirs of the American Mathematical Society, ISSN 0065-9266, E-ISSN 1947-6221, Vol. 218, no 1024, 1-+ p.Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2012. Vol. 218, no 1024, 1-+ p.
##### Keyword [en]

Pure O-sequence, Artinian algebra, monomial algebra, unimodality, differentiable O-sequence, level algebra, Gorenstein algebra, enumeration, interval conjecture, g-element, weak Lefschetz property, strong Lefschetz property, matroid sirnplicial complex, Macaulay's inverse system
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:kth:diva-99067ISI: 000305504300001OAI: oai:DiVA.org:kth-99067DiVA: diva2:541558
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt434",{id:"formSmash:j_idt434",widgetVar:"widget_formSmash_j_idt434",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt440",{id:"formSmash:j_idt440",widgetVar:"widget_formSmash_j_idt440",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt446",{id:"formSmash:j_idt446",widgetVar:"widget_formSmash_j_idt446",multiple:true});
##### Note

QC 20120719Available from: 2012-07-19 Created: 2012-07-13 Last updated: 2017-12-07Bibliographically approved

A monomial order ideal is a finite collection X of (monic) monomials such that, whenever M is an element of X and N divides M, then N is an element of X. Hence X is a poset, where the partial order is given by divisibility. If all, say t, maximal monomials of X have the same degree, then X is pure (of type t). A pure O-sequence is the vector, (h) under bar = (h(0) = 1, h(1), ..., h(e)), counting the monomials of X in each degree. Equivalently, pure O-sequences can be characterized as the f-vectors of pure multicomplexes, or, in the language of commutative algebra, as the h-vectors of monomial Artinian level algebras. Pure O-sequences had their origin in one of the early works of Stanley's in this area, and have since played a significant role in at least three different disciplines: the study of simplicial complexes and their f-vectors, the theory of level algebras, and the theory of matroids. This monograph is intended to be the first systematic study of the theory of pure O-sequences. Our work, which makes an extensive use of both algebraic and combinatorial techniques, in particular includes: (i) A characterization of the first half of a pure O-sequence, which yields the exact converse to a g-theorem of Hausel; (ii) A study of (the failing of) the unimodality property; (iii) The problem of enumerating pure O-sequences, including a proof that almost all O-sequences are pure, a natural bijection between integer partitions and type 1 pure O-sequences, and the asymptotic enumeration of socle degree 3 pure O-sequences of type t; (iv) A study of the Interval Conjecture for Pure O-sequences (ICP), which represents perhaps the strongest possible structural result short of an (impossible?) full characterization; (v) A pithy connection of the ICP with Stanley's conjecture on the h-vectors of matroid complexes; (vi) A more specific study of pure O-sequences of type 2, including a proof of the Weak Lefschetz Property in codimension 3 over a field of characteristic zero. As an immediate corollary, pure O-sequences of codimension 3 and type 2 are unimodal (over an arbitrary field). (vii) An analysis, from a commutative algebra viewpoint, of the extent to which the Weak and Strong Lefschetz Properties can fail for monomial algebras. (viii) Some observations about pure f-vectors, an important special case of pure O-sequences.

urn-nbn$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_j_idt1144",{id:"formSmash:j_idt1144",widgetVar:"widget_formSmash_j_idt1144",showEffect:"fade",hideEffect:"fade",showDelay:500,hideDelay:300,target:"formSmash:altmetricDiv"});});

CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1197",{id:"formSmash:lower:j_idt1197",widgetVar:"widget_formSmash_lower_j_idt1197",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1198_j_idt1200",{id:"formSmash:lower:j_idt1198:j_idt1200",widgetVar:"widget_formSmash_lower_j_idt1198_j_idt1200",target:"formSmash:lower:j_idt1198:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});