Skip to main content Accessibility help
×
Home

THE EXPECTED JAGGEDNESS OF ORDER IDEALS

  • MELODY CHAN (a1), SHAHRZAD HADDADAN (a2), SAM HOPKINS (a3) and LUCA MOCI (a4)

Abstract

The jaggedness of an order ideal $I$ in a poset $P$ is the number of maximal elements in $I$ plus the number of minimal elements of $P$ not in $I$ . A probability distribution on the set of order ideals of $P$ is toggle-symmetric if for every $p\in P$ , the probability that $p$ is maximal in $I$ equals the probability that $p$ is minimal not in $I$ . In this paper, we prove a formula for the expected jaggedness of an order ideal of  $P$ under any toggle-symmetric probability distribution when $P$ is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan–López–Pflueger–Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp–Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      THE EXPECTED JAGGEDNESS OF ORDER IDEALS
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      THE EXPECTED JAGGEDNESS OF ORDER IDEALS
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      THE EXPECTED JAGGEDNESS OF ORDER IDEALS
      Available formats
      ×

Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

Hide All
[1] Aitken, A. C., ‘The monomial expansion of determinantal symmetric functions’, Proc. Roy. Soc. Edinburgh. Sect. A 61 (1943), 300310.
[2] Ayyer, A., Schilling, A. and Thiéry, N. M., ‘Spectral gap for random-to-random shuffling on linear extensions’, Exp. Math. 26(1) (2017), 2230.
[3] Bloom, J., Pechenik, O. and Saracino, D., ‘Proofs and generalizations of a homomesy conjecture of Propp and Roby’, Discrete Math. 339(1) (2016), 194206.
[4] Brouwer, A. E. and Schrijver, A., On the period of an operator, defined on antichains. Mathematisch Centrum, Amsterdam, 1974. Mathematisch Centrum Afdeling Zuivere Wiskunde ZW 24/74.
[5] Cameron, P. J. and Fon-Der-Flaass, D. G., ‘Orbits of antichains revisited’, European J. Combin. 16(6) (1995), 545554.
[6] Chan, M., López Martín, A., Pflueger, N. and Teixidor i Bigas, M., ‘Genera of Brill-Noether curves and staircase paths in Young tableaux’, Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516.
[7] Einstein, D., Farber, M., Gunawan, E., Joseph, M., Macauley, M., Propp, J. and Rubinstein-Salzedo, S., ‘Noncrossing partitions, toggles, and homomesies’, Preprint, 2015, arXiv:1510.06362.
[8] Eisenbud, D. and Harris, J., ‘Limit linear series: basic theory’, Invent. Math. 85(2) (1986), 337371.
[9] Gessel, I. M., ‘A historical survey of P-partitions’, inThe Mathematical Legacy of Richard P. Stanley (eds. Hersh, P., Lam, T., Pylyavskyy, P. and Reiner, V.) (American Mathematical Society, Providence, RI, 2016), 169188.
[10] Kreweras, G., ‘Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers’, Cah. BURO 6 (1965), 9107.
[11] MacMahon, P. A., Combinatory Analysis, Dover Phoenix Editions, Vol. I, II (bound in one volume) (Dover Publications, Inc., Mineola, NY, 2004), Reprint of An Introduction to Combinatory Analysis (1920) and Combinatory Analysis. Vol. I, II (1915, 1916).
[12] Propp, J. and Roby, T., ‘Homomesy in the product of two chains’, Electron. J. Combin. 22(3) (2015), Paper 3, 4.
[13] Reiner, V., Stanton, D. and White, D., ‘What is … cyclic sieving?’, Notices Amer. Math. Soc. 61(2) (2014), 169171.
[14] Roby, T., ‘Dynamical algebraic combinatorics and the homomesy phenomenon’, inRecent Trends in Combinatorics, The IMA Volumes in Mathematics and its Applications, 159 (Springer International Publishing Switzerland, 2016), 619652. [Cham].
[15] Stanley, R. P., Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, 62 (Cambridge University Press, Cambridge, 1999).
[16] Stanley, R. P., Enumerative Combinatorics, 2nd edn, Vol. 1, Cambridge Studies in Advanced Mathematics, 49 (Cambridge University Press, Cambridge, 2012).
[17] Striker, J., ‘The toggle group, homomesy, and the Razumov–Stroganov correspondence’, Electron. J. Combin. 22(2) (2015), Paper 2, 57.
[18] Striker, J. and Williams, N., ‘Promotion and rowmotion’, European J. Combin. 33(8) (2012), 19191942.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

THE EXPECTED JAGGEDNESS OF ORDER IDEALS

  • MELODY CHAN (a1), SHAHRZAD HADDADAN (a2), SAM HOPKINS (a3) and LUCA MOCI (a4)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed