## Abstract

In certain finite posets, the expected down-degree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it. We show that this coincidence of expectations holds for Cartesian products of chains, connected minuscule posets, weak Bruhat orders on finite Coxeter groups, certain lower intervals in Young's lattice, and certain lower intervals in the weak Bruhat order below dominant permutations. Our tools involve formulas for counting nearly reduced factorizations in 0-Hecke algebras; that is, factorizations that are one letter longer than the Coxeter group length.

Original language | English (US) |
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Pages (from-to) | 66-125 |

Number of pages | 60 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 158 |

DOIs | |

State | Published - Aug 2018 |

### Bibliographical note

Funding Information:The authors thank Zach Hamaker for pointing out an earlier argument for Lemma 5.4 in terms of Lascoux's transition formula for Grothendieck polynomials [27] . We thank Kári Ragnarsson and Cara Monical for providing computer code useful in the investigations of Sections 2 and 6 , respectively. We thank Susanna Fishel and Thomas McConville for pointing out Proposition 2.20 . We thank Thomas McConville for pointing out the examples at the end of Section 2.2.2 , and thank Alex Garver for help on oriented exchange graphs. We thank Sam Hopkins for helpful discussions on the work in [9] , and for sharing with us his preliminary results on shifted Young diagrams. We thank Travis Scrimshaw for pointing out the connection between uncrowding and [6, Theorem 6.11] (specifically the proof sketch). We also thank Bruce Sagan for several helpful comments and questions. Finally, we thank the anonymous referees for their suggestions. VR was partially supported by National Science Foundation RTG grant DMS-1148634 . BT was partially supported by a Simons Foundation Collaboration Grant for Mathematicians 277603 . AY was partially supported by National Science Foundation grant DMS-1500691 .

Funding Information:

The authors thank Zach Hamaker for pointing out an earlier argument for Lemma 5.4 in terms of Lascoux's transition formula for Grothendieck polynomials [27]. We thank Kári Ragnarsson and Cara Monical for providing computer code useful in the investigations of Sections 2 and 6, respectively. We thank Susanna Fishel and Thomas McConville for pointing out Proposition 2.20. We thank Thomas McConville for pointing out the examples at the end of Section 2.2.2, and thank Alex Garver for help on oriented exchange graphs. We thank Sam Hopkins for helpful discussions on the work in [9], and for sharing with us his preliminary results on shifted Young diagrams. We thank Travis Scrimshaw for pointing out the connection between uncrowding and [6, Theorem 6.11] (specifically the proof sketch). We also thank Bruce Sagan for several helpful comments and questions. Finally, we thank the anonymous referees for their suggestions. VR was partially supported by National Science Foundation RTG grant DMS-1148634. BT was partially supported by a Simons Foundation Collaboration Grant for Mathematicians 277603. AY was partially supported by National Science Foundation grant DMS-1500691.

Publisher Copyright:

© 2018 Elsevier Inc.

## Keywords

- 0-Hecke
- Dominant
- Grothendieck polynomial
- Monoid
- NilHecke
- Rectangular shape
- Reduced word
- Set-valued
- Staircase shape
- Tableau