# Enumeration of permutations by rises, falls, rising maxima and falling maxima

Enumeration of permutations by rises, falls, rising maxima and falling maxima Acta 3([athematica Academiae Scientiarum Hungaricae Tomrs 25 (3--4), (1974), pp. 269--277. ENUMERATION OF PERMUTATIONS BY RISES, FALLS, RISING MAXIMA AND FALLING MAXIMA By L. CARLITZ and R. SCOVILLE (Durham) Let A(r, s) denote the number of permutations with r+l rises and s+l falls, where a conventional rise on the left and a conventional fall on the right are counted. It is known that [2] xr y s e x _ e y Z A(r,s) - - F(x,y). xeY-- ye x (r+s+ 1)! r~s=O Moreover if P(r, s, k) denotes the number of permutations with r rises, s falls and k maxima, it has been proved that rain @,s) Xr jS Z k X ~ P(r+l,,~+l,k+l) -F(U,V), .... o ~=0 (r+s+ 1)t where 1 [x+y+l/(x+y)~_4xyz] ' V=-~ 1 [x +y - V(x +y)2_4xyz]. U=-s In the present paper, recurrences and generating functions are obtained for the number of permutations with a given number of rises, fails, rising maxima and falling maxima. 1. Let (1.1) rc = (al, a2 ..... a,) denote an arbitrary permutation of Z,= {1, 2, ..., n}. By a rise is meant a pair of consecutive elements ai, ai+l with ai<ai+~. By a fail is meant a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

# Enumeration of permutations by rises, falls, rising maxima and falling maxima

, Volume 25 (4) – Sep 1, 1974
9 pages

/lp/springer-journals/enumeration-of-permutations-by-rises-falls-rising-maxima-and-falling-ij4aAYwRGZ
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01886084
Publisher site
See Article on Publisher Site

### Abstract

Acta 3([athematica Academiae Scientiarum Hungaricae Tomrs 25 (3--4), (1974), pp. 269--277. ENUMERATION OF PERMUTATIONS BY RISES, FALLS, RISING MAXIMA AND FALLING MAXIMA By L. CARLITZ and R. SCOVILLE (Durham) Let A(r, s) denote the number of permutations with r+l rises and s+l falls, where a conventional rise on the left and a conventional fall on the right are counted. It is known that [2] xr y s e x _ e y Z A(r,s) - - F(x,y). xeY-- ye x (r+s+ 1)! r~s=O Moreover if P(r, s, k) denotes the number of permutations with r rises, s falls and k maxima, it has been proved that rain @,s) Xr jS Z k X ~ P(r+l,,~+l,k+l) -F(U,V), .... o ~=0 (r+s+ 1)t where 1 [x+y+l/(x+y)~_4xyz] ' V=-~ 1 [x +y - V(x +y)2_4xyz]. U=-s In the present paper, recurrences and generating functions are obtained for the number of permutations with a given number of rises, fails, rising maxima and falling maxima. 1. Let (1.1) rc = (al, a2 ..... a,) denote an arbitrary permutation of Z,= {1, 2, ..., n}. By a rise is meant a pair of consecutive elements ai, ai+l with ai<ai+~. By a fail is meant a

### Journal

Acta Mathematica Academiae Scientiarum HungaricaSpringer Journals

Published: Sep 1, 1974

### References

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