Connecting descent and peak polynomials


Creative Commons License

Oğuz E. K.

Hacettepe Journal of Mathematics and Statistics, cilt.53, sa.2, ss.488-494, 2024 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 53 Sayı: 2
  • Basım Tarihi: 2024
  • Doi Numarası: 10.15672/hujms.1182500
  • Dergi Adı: Hacettepe Journal of Mathematics and Statistics
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, zbMATH
  • Sayfa Sayıları: ss.488-494
  • Anahtar Kelimeler: binomial coefficients, descent, peak, permutations
  • Galatasaray Üniversitesi Adresli: Evet

Özet

A permutation σ = σ1 σ2 ... σn has a descent at i if σi > σi+1. A descent i is called a peak if i > 1 and i − 1 is not a descent. The size of the set of all permutations of n with a given descent set is a polynomials in n, called the descent polynomial. Similarly, the size of the set of all permutations of n with a given peak set, adjusted by a power of 2 gives a polynomial in n, called the peak polynomial. In this work we give a unitary expansion of descent polynomials in terms of peak polynomials. Then we use this expansion to give an interpretation of the coefficients of the peak polynomial in a binomial basis, thus giving a constructive proof of the peak polynomial positivity conjecture.