A grammatical calculus for peaks and runs of permutations

Value-Peaks of Permutations

In this paper, we focus on a “local property” of permutations: value-peak. A permutation σ has a value-peak σ(i) if σ(i − 1) < σ(i) > σ(i + 1) for some i ∈ [2, n − 1]. Define V P (σ) as the set of value-peaks of the permutation σ. For any S ⊆ [3, n], define V Pn(S) such that V P (σ) = S. Let Pn = {S | V Pn(S) 6= ∅}. we make the set Pn into a poset Pn by defining S T if S ⊆ T as sets. We prove t...

the innovation of a statistical model to estimate dependable rainfall (dr) and develop it for determination and classification of drought and wet years of iran

آب حاصل از بارش منبع تأمین نیازهای بی شمار جانداران به ویژه انسان است و هرگونه کاهش در کم و کیف آن مستقیماً حیات موجودات زنده را تحت تأثیر منفی قرار می دهد. نوسان سال به سال بارش از ویژگی های اساسی و بسیار مهم بارش های سالانه ایران محسوب می شود که آثار زیان بار آن در تمام عرصه های اقتصادی، اجتماعی و حتی سیاسی- امنیتی به نحوی منعکس می شود. چون میزان آب ناشی از بارش یکی از مولفه های اصلی برنامه ...

Counting permutations by their alternating runs

We find a formula for the number of permutations of [n] that have exactly s runs up and down. The formula is at once terminating, asymptotic, and exact. The asymptotic series is valid for n→∞, uniformly for s (1 − )n/ logn ( > 0). © 2007 Elsevier Inc. All rights reserved.

2-stack Sortable Permutations with a given Number of Runs

Using earlier results we prove a formula for the number W(n,k) of 2stack sortable permutations of length n with k runs, or in other words, k − 1 descents. This formula will yield the suprising fact that there are as many 2-stack sortable permutations with k−1 descents as with k−1 ascents. We also prove that W(n,k) is unimodal in k, for any fixed n.

Counting Permutations by Their Runs Up and DownE