Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

Crossings and Nestings of Matchings and Partitions

William Y. C. Chen, Eva Y. P. Deng, Rosena R. X. Du, Richard P. Stanley and Catherine H. Yan
Transactions of the American Mathematical Society
Vol. 359, No. 4 (Apr., 2007), pp. 1555-1575
Stable URL: http://www.jstor.org/stable/20161642
Page Count: 21
  • Read Online (Free)
  • Download ($30.00)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Crossings and Nestings of Matchings and Partitions
Preview not available

Abstract

We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of [n], as well as over all matchings on [2n]. As a corollary, the number of k-noncrossing partitions is equal to the number of k-nonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no k-crossing (or with no k-nesting).

Page Thumbnails

  • Thumbnail: Page 
1555
    1555
  • Thumbnail: Page 
1556
    1556
  • Thumbnail: Page 
1557
    1557
  • Thumbnail: Page 
1558
    1558
  • Thumbnail: Page 
1559
    1559
  • Thumbnail: Page 
1560
    1560
  • Thumbnail: Page 
1561
    1561
  • Thumbnail: Page 
1562
    1562
  • Thumbnail: Page 
1563
    1563
  • Thumbnail: Page 
1564
    1564
  • Thumbnail: Page 
1565
    1565
  • Thumbnail: Page 
1566
    1566
  • Thumbnail: Page 
1567
    1567
  • Thumbnail: Page 
1568
    1568
  • Thumbnail: Page 
1569
    1569
  • Thumbnail: Page 
1570
    1570
  • Thumbnail: Page 
1571
    1571
  • Thumbnail: Page 
1572
    1572
  • Thumbnail: Page 
1573
    1573
  • Thumbnail: Page 
1574
    1574
  • Thumbnail: Page 
1575
    1575