Abstract
We prove two conjectures of Shareshian and Wachs about Eulerian quasisymmetric functions and Eulerian polynomials. The first states that the cycle type Eulerian quasisymmetric function Qλ,j is Schur-positive, and moreover that the sequence Qλ,j as j varies is Schur-unimodal. The second conjecture, which we prove using the first, states that the cycle type (q, p)-Eulerian polynomial Aλmaj,des,exc(q,p,q-1t) is t-unimodal.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 135-145 |
| Number of pages | 11 |
| Journal | Journal of Combinatorial Theory, Series A |
| Volume | 119 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jan 2012 |
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Keywords
- Cycle type
- Eulerian polynomials
- Permutation statistics
- Symmetric functions
- Unimodality
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Theoretical Computer Science
- Computational Theory and Mathematics
Cite this
Unimodality of Eulerian quasisymmetric functions. / Henderson, Anthony; Galloway, Michelle L.
In: Journal of Combinatorial Theory, Series A, Vol. 119, No. 1, 01.2012, p. 135-145.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Unimodality of Eulerian quasisymmetric functions
AU - Henderson, Anthony
AU - Galloway, Michelle L
PY - 2012/1
Y1 - 2012/1
N2 - We prove two conjectures of Shareshian and Wachs about Eulerian quasisymmetric functions and Eulerian polynomials. The first states that the cycle type Eulerian quasisymmetric function Qλ,j is Schur-positive, and moreover that the sequence Qλ,j as j varies is Schur-unimodal. The second conjecture, which we prove using the first, states that the cycle type (q, p)-Eulerian polynomial Aλmaj,des,exc(q,p,q-1t) is t-unimodal.
AB - We prove two conjectures of Shareshian and Wachs about Eulerian quasisymmetric functions and Eulerian polynomials. The first states that the cycle type Eulerian quasisymmetric function Qλ,j is Schur-positive, and moreover that the sequence Qλ,j as j varies is Schur-unimodal. The second conjecture, which we prove using the first, states that the cycle type (q, p)-Eulerian polynomial Aλmaj,des,exc(q,p,q-1t) is t-unimodal.
KW - Cycle type
KW - Eulerian polynomials
KW - Permutation statistics
KW - Symmetric functions
KW - Unimodality
UR - http://www.scopus.com/inward/record.url?scp=80052092606&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=80052092606&partnerID=8YFLogxK
U2 - 10.1016/j.jcta.2011.07.004
DO - 10.1016/j.jcta.2011.07.004
M3 - Article
AN - SCOPUS:80052092606
VL - 119
SP - 135
EP - 145
JO - Journal of Combinatorial Theory - Series A
JF - Journal of Combinatorial Theory - Series A
SN - 0097-3165
IS - 1
ER -