*****************************************************************************
*       www.FindStat.org - The Combinatorial Statistic Finder               *
*                                                                           *
*       Copyright (C) 2019 The FindStatCrew <info@findstat.org>             *
*                                                                           *
*    This information is distributed in the hope that it will be useful,    *
*    but WITHOUT ANY WARRANTY; without even the implied warranty of         *
*    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.                   *
*****************************************************************************

-----------------------------------------------------------------------------
Statistic identifier: St000761

-----------------------------------------------------------------------------
Collection: Integer compositions

-----------------------------------------------------------------------------
Description: The number of ascents in an integer composition.

A composition has an ascent, or rise, at position $i$ if $a_i < a_{i+1}$.

-----------------------------------------------------------------------------
References: [1]   Carlitz, L. Enumeration of compositions by rises, falls and levels [[MathSciNet:0450078]]

-----------------------------------------------------------------------------
Code:
def statistic(pi):
    res = 0
    for i in range(len(pi)-1):
        if pi[i] < pi[i+1]:
            res += 1
    return res


-----------------------------------------------------------------------------
Statistic values:

[1]                 => 0
[1,1]               => 0
[2]                 => 0
[1,1,1]             => 0
[1,2]               => 1
[2,1]               => 0
[3]                 => 0
[1,1,1,1]           => 0
[1,1,2]             => 1
[1,2,1]             => 1
[1,3]               => 1
[2,1,1]             => 0
[2,2]               => 0
[3,1]               => 0
[4]                 => 0
[1,1,1,1,1]         => 0
[1,1,1,2]           => 1
[1,1,2,1]           => 1
[1,1,3]             => 1
[1,2,1,1]           => 1
[1,2,2]             => 1
[1,3,1]             => 1
[1,4]               => 1
[2,1,1,1]           => 0
[2,1,2]             => 1
[2,2,1]             => 0
[2,3]               => 1
[3,1,1]             => 0
[3,2]               => 0
[4,1]               => 0
[5]                 => 0
[1,1,1,1,1,1]       => 0
[1,1,1,1,2]         => 1
[1,1,1,2,1]         => 1
[1,1,1,3]           => 1
[1,1,2,1,1]         => 1
[1,1,2,2]           => 1
[1,1,3,1]           => 1
[1,1,4]             => 1
[1,2,1,1,1]         => 1
[1,2,1,2]           => 2
[1,2,2,1]           => 1
[1,2,3]             => 2
[1,3,1,1]           => 1
[1,3,2]             => 1
[1,4,1]             => 1
[1,5]               => 1
[2,1,1,1,1]         => 0
[2,1,1,2]           => 1
[2,1,2,1]           => 1
[2,1,3]             => 1
[2,2,1,1]           => 0
[2,2,2]             => 0
[2,3,1]             => 1
[2,4]               => 1
[3,1,1,1]           => 0
[3,1,2]             => 1
[3,2,1]             => 0
[3,3]               => 0
[4,1,1]             => 0
[4,2]               => 0
[5,1]               => 0
[6]                 => 0
[1,1,1,1,1,1,1]     => 0
[1,1,1,1,1,2]       => 1
[1,1,1,1,2,1]       => 1
[1,1,1,1,3]         => 1
[1,1,1,2,1,1]       => 1
[1,1,1,2,2]         => 1
[1,1,1,3,1]         => 1
[1,1,1,4]           => 1
[1,1,2,1,1,1]       => 1
[1,1,2,1,2]         => 2
[1,1,2,2,1]         => 1
[1,1,2,3]           => 2
[1,1,3,1,1]         => 1
[1,1,3,2]           => 1
[1,1,4,1]           => 1
[1,1,5]             => 1
[1,2,1,1,1,1]       => 1
[1,2,1,1,2]         => 2
[1,2,1,2,1]         => 2
[1,2,1,3]           => 2
[1,2,2,1,1]         => 1
[1,2,2,2]           => 1
[1,2,3,1]           => 2
[1,2,4]             => 2
[1,3,1,1,1]         => 1
[1,3,1,2]           => 2
[1,3,2,1]           => 1
[1,3,3]             => 1
[1,4,1,1]           => 1
[1,4,2]             => 1
[1,5,1]             => 1
[1,6]               => 1
[2,1,1,1,1,1]       => 0
[2,1,1,1,2]         => 1
[2,1,1,2,1]         => 1
[2,1,1,3]           => 1
[2,1,2,1,1]         => 1
[2,1,2,2]           => 1
[2,1,3,1]           => 1
[2,1,4]             => 1
[2,2,1,1,1]         => 0
[2,2,1,2]           => 1
[2,2,2,1]           => 0
[2,2,3]             => 1
[2,3,1,1]           => 1
[2,3,2]             => 1
[2,4,1]             => 1
[2,5]               => 1
[3,1,1,1,1]         => 0
[3,1,1,2]           => 1
[3,1,2,1]           => 1
[3,1,3]             => 1
[3,2,1,1]           => 0
[3,2,2]             => 0
[3,3,1]             => 0
[3,4]               => 1
[4,1,1,1]           => 0
[4,1,2]             => 1
[4,2,1]             => 0
[4,3]               => 0
[5,1,1]             => 0
[5,2]               => 0
[6,1]               => 0
[7]                 => 0
[1,1,1,1,1,1,1,1]   => 0
[1,1,1,1,1,1,2]     => 1
[1,1,1,1,1,2,1]     => 1
[1,1,1,1,1,3]       => 1
[1,1,1,1,2,1,1]     => 1
[1,1,1,1,2,2]       => 1
[1,1,1,1,3,1]       => 1
[1,1,1,1,4]         => 1
[1,1,1,2,1,1,1]     => 1
[1,1,1,2,1,2]       => 2
[1,1,1,2,2,1]       => 1
[1,1,1,2,3]         => 2
[1,1,1,3,1,1]       => 1
[1,1,1,3,2]         => 1
[1,1,1,4,1]         => 1
[1,1,1,5]           => 1
[1,1,2,1,1,1,1]     => 1
[1,1,2,1,1,2]       => 2
[1,1,2,1,2,1]       => 2
[1,1,2,1,3]         => 2
[1,1,2,2,1,1]       => 1
[1,1,2,2,2]         => 1
[1,1,2,3,1]         => 2
[1,1,2,4]           => 2
[1,1,3,1,1,1]       => 1
[1,1,3,1,2]         => 2
[1,1,3,2,1]         => 1
[1,1,3,3]           => 1
[1,1,4,1,1]         => 1
[1,1,4,2]           => 1
[1,1,5,1]           => 1
[1,1,6]             => 1
[1,2,1,1,1,1,1]     => 1
[1,2,1,1,1,2]       => 2
[1,2,1,1,2,1]       => 2
[1,2,1,1,3]         => 2
[1,2,1,2,1,1]       => 2
[1,2,1,2,2]         => 2
[1,2,1,3,1]         => 2
[1,2,1,4]           => 2
[1,2,2,1,1,1]       => 1
[1,2,2,1,2]         => 2
[1,2,2,2,1]         => 1
[1,2,2,3]           => 2
[1,2,3,1,1]         => 2
[1,2,3,2]           => 2
[1,2,4,1]           => 2
[1,2,5]             => 2
[1,3,1,1,1,1]       => 1
[1,3,1,1,2]         => 2
[1,3,1,2,1]         => 2
[1,3,1,3]           => 2
[1,3,2,1,1]         => 1
[1,3,2,2]           => 1
[1,3,3,1]           => 1
[1,3,4]             => 2
[1,4,1,1,1]         => 1
[1,4,1,2]           => 2
[1,4,2,1]           => 1
[1,4,3]             => 1
[1,5,1,1]           => 1
[1,5,2]             => 1
[1,6,1]             => 1
[1,7]               => 1
[2,1,1,1,1,1,1]     => 0
[2,1,1,1,1,2]       => 1
[2,1,1,1,2,1]       => 1
[2,1,1,1,3]         => 1
[2,1,1,2,1,1]       => 1
[2,1,1,2,2]         => 1
[2,1,1,3,1]         => 1
[2,1,1,4]           => 1
[2,1,2,1,1,1]       => 1
[2,1,2,1,2]         => 2
[2,1,2,2,1]         => 1
[2,1,2,3]           => 2
[2,1,3,1,1]         => 1
[2,1,3,2]           => 1
[2,1,4,1]           => 1
[2,1,5]             => 1
[2,2,1,1,1,1]       => 0
[2,2,1,1,2]         => 1
[2,2,1,2,1]         => 1
[2,2,1,3]           => 1
[2,2,2,1,1]         => 0
[2,2,2,2]           => 0
[2,2,3,1]           => 1
[2,2,4]             => 1
[2,3,1,1,1]         => 1
[2,3,1,2]           => 2
[2,3,2,1]           => 1
[2,3,3]             => 1
[2,4,1,1]           => 1
[2,4,2]             => 1
[2,5,1]             => 1
[2,6]               => 1
[3,1,1,1,1,1]       => 0
[3,1,1,1,2]         => 1
[3,1,1,2,1]         => 1
[3,1,1,3]           => 1
[3,1,2,1,1]         => 1
[3,1,2,2]           => 1
[3,1,3,1]           => 1
[3,1,4]             => 1
[3,2,1,1,1]         => 0
[3,2,1,2]           => 1
[3,2,2,1]           => 0
[3,2,3]             => 1
[3,3,1,1]           => 0
[3,3,2]             => 0
[3,4,1]             => 1
[3,5]               => 1
[4,1,1,1,1]         => 0
[4,1,1,2]           => 1
[4,1,2,1]           => 1
[4,1,3]             => 1
[4,2,1,1]           => 0
[4,2,2]             => 0
[4,3,1]             => 0
[4,4]               => 0
[5,1,1,1]           => 0
[5,1,2]             => 1
[5,2,1]             => 0
[5,3]               => 0
[6,1,1]             => 0
[6,2]               => 0
[7,1]               => 0
[8]                 => 0
[1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,2]   => 1
[1,1,1,1,1,1,2,1]   => 1
[1,1,1,1,1,1,3]     => 1
[1,1,1,1,1,2,1,1]   => 1
[1,1,1,1,1,2,2]     => 1
[1,1,1,1,1,3,1]     => 1
[1,1,1,1,1,4]       => 1
[1,1,1,1,2,1,1,1]   => 1
[1,1,1,1,2,1,2]     => 2
[1,1,1,1,2,2,1]     => 1
[1,1,1,1,2,3]       => 2
[1,1,1,1,3,1,1]     => 1
[1,1,1,1,3,2]       => 1
[1,1,1,1,4,1]       => 1
[1,1,1,1,5]         => 1
[1,1,1,2,1,1,1,1]   => 1
[1,1,1,2,1,1,2]     => 2
[1,1,1,2,1,2,1]     => 2
[1,1,1,2,1,3]       => 2
[1,1,1,2,2,1,1]     => 1
[1,1,1,2,2,2]       => 1
[1,1,1,2,3,1]       => 2
[1,1,1,2,4]         => 2
[1,1,1,3,1,1,1]     => 1
[1,1,1,3,1,2]       => 2
[1,1,1,3,2,1]       => 1
[1,1,1,3,3]         => 1
[1,1,1,4,1,1]       => 1
[1,1,1,4,2]         => 1
[1,1,1,5,1]         => 1
[1,1,1,6]           => 1
[1,1,2,1,1,1,1,1]   => 1
[1,1,2,1,1,1,2]     => 2
[1,1,2,1,1,2,1]     => 2
[1,1,2,1,1,3]       => 2
[1,1,2,1,2,1,1]     => 2
[1,1,2,1,2,2]       => 2
[1,1,2,1,3,1]       => 2
[1,1,2,1,4]         => 2
[1,1,2,2,1,1,1]     => 1
[1,1,2,2,1,2]       => 2
[1,1,2,2,2,1]       => 1
[1,1,2,2,3]         => 2
[1,1,2,3,1,1]       => 2
[1,1,2,3,2]         => 2
[1,1,2,4,1]         => 2
[1,1,2,5]           => 2
[1,1,3,1,1,1,1]     => 1
[1,1,3,1,1,2]       => 2
[1,1,3,1,2,1]       => 2
[1,1,3,1,3]         => 2
[1,1,3,2,1,1]       => 1
[1,1,3,2,2]         => 1
[1,1,3,3,1]         => 1
[1,1,3,4]           => 2
[1,1,4,1,1,1]       => 1
[1,1,4,1,2]         => 2
[1,1,4,2,1]         => 1
[1,1,4,3]           => 1
[1,1,5,1,1]         => 1
[1,1,5,2]           => 1
[1,1,6,1]           => 1
[1,1,7]             => 1
[1,2,1,1,1,1,1,1]   => 1
[1,2,1,1,1,1,2]     => 2
[1,2,1,1,1,2,1]     => 2
[1,2,1,1,1,3]       => 2
[1,2,1,1,2,1,1]     => 2
[1,2,1,1,2,2]       => 2
[1,2,1,1,3,1]       => 2
[1,2,1,1,4]         => 2
[1,2,1,2,1,1,1]     => 2
[1,2,1,2,1,2]       => 3
[1,2,1,2,2,1]       => 2
[1,2,1,2,3]         => 3
[1,2,1,3,1,1]       => 2
[1,2,1,3,2]         => 2
[1,2,1,4,1]         => 2
[1,2,1,5]           => 2
[1,2,2,1,1,1,1]     => 1
[1,2,2,1,1,2]       => 2
[1,2,2,1,2,1]       => 2
[1,2,2,1,3]         => 2
[1,2,2,2,1,1]       => 1
[1,2,2,2,2]         => 1
[1,2,2,3,1]         => 2
[1,2,2,4]           => 2
[1,2,3,1,1,1]       => 2
[1,2,3,1,2]         => 3
[1,2,3,2,1]         => 2
[1,2,3,3]           => 2
[1,2,4,1,1]         => 2
[1,2,4,2]           => 2
[1,2,5,1]           => 2
[1,2,6]             => 2
[1,3,1,1,1,1,1]     => 1
[1,3,1,1,1,2]       => 2
[1,3,1,1,2,1]       => 2
[1,3,1,1,3]         => 2
[1,3,1,2,1,1]       => 2
[1,3,1,2,2]         => 2
[1,3,1,3,1]         => 2
[1,3,1,4]           => 2
[1,3,2,1,1,1]       => 1
[1,3,2,1,2]         => 2
[1,3,2,2,1]         => 1
[1,3,2,3]           => 2
[1,3,3,1,1]         => 1
[1,3,3,2]           => 1
[1,3,4,1]           => 2
[1,3,5]             => 2
[1,4,1,1,1,1]       => 1
[1,4,1,1,2]         => 2
[1,4,1,2,1]         => 2
[1,4,1,3]           => 2
[1,4,2,1,1]         => 1
[1,4,2,2]           => 1
[1,4,3,1]           => 1
[1,4,4]             => 1
[1,5,1,1,1]         => 1
[1,5,1,2]           => 2
[1,5,2,1]           => 1
[1,5,3]             => 1
[1,6,1,1]           => 1
[1,6,2]             => 1
[1,7,1]             => 1
[1,8]               => 1
[2,1,1,1,1,1,1,1]   => 0
[2,1,1,1,1,1,2]     => 1
[2,1,1,1,1,2,1]     => 1
[2,1,1,1,1,3]       => 1
[2,1,1,1,2,1,1]     => 1
[2,1,1,1,2,2]       => 1
[2,1,1,1,3,1]       => 1
[2,1,1,1,4]         => 1
[2,1,1,2,1,1,1]     => 1
[2,1,1,2,1,2]       => 2
[2,1,1,2,2,1]       => 1
[2,1,1,2,3]         => 2
[2,1,1,3,1,1]       => 1
[2,1,1,3,2]         => 1
[2,1,1,4,1]         => 1
[2,1,1,5]           => 1
[2,1,2,1,1,1,1]     => 1
[2,1,2,1,1,2]       => 2
[2,1,2,1,2,1]       => 2
[2,1,2,1,3]         => 2
[2,1,2,2,1,1]       => 1
[2,1,2,2,2]         => 1
[2,1,2,3,1]         => 2
[2,1,2,4]           => 2
[2,1,3,1,1,1]       => 1
[2,1,3,1,2]         => 2
[2,1,3,2,1]         => 1
[2,1,3,3]           => 1
[2,1,4,1,1]         => 1
[2,1,4,2]           => 1
[2,1,5,1]           => 1
[2,1,6]             => 1
[2,2,1,1,1,1,1]     => 0
[2,2,1,1,1,2]       => 1
[2,2,1,1,2,1]       => 1
[2,2,1,1,3]         => 1
[2,2,1,2,1,1]       => 1
[2,2,1,2,2]         => 1
[2,2,1,3,1]         => 1
[2,2,1,4]           => 1
[2,2,2,1,1,1]       => 0
[2,2,2,1,2]         => 1
[2,2,2,2,1]         => 0
[2,2,2,3]           => 1
[2,2,3,1,1]         => 1
[2,2,3,2]           => 1
[2,2,4,1]           => 1
[2,2,5]             => 1
[2,3,1,1,1,1]       => 1
[2,3,1,1,2]         => 2
[2,3,1,2,1]         => 2
[2,3,1,3]           => 2
[2,3,2,1,1]         => 1
[2,3,2,2]           => 1
[2,3,3,1]           => 1
[2,3,4]             => 2
[2,4,1,1,1]         => 1
[2,4,1,2]           => 2
[2,4,2,1]           => 1
[2,4,3]             => 1
[2,5,1,1]           => 1
[2,5,2]             => 1
[2,6,1]             => 1
[2,7]               => 1
[3,1,1,1,1,1,1]     => 0
[3,1,1,1,1,2]       => 1
[3,1,1,1,2,1]       => 1
[3,1,1,1,3]         => 1
[3,1,1,2,1,1]       => 1
[3,1,1,2,2]         => 1
[3,1,1,3,1]         => 1
[3,1,1,4]           => 1
[3,1,2,1,1,1]       => 1
[3,1,2,1,2]         => 2
[3,1,2,2,1]         => 1
[3,1,2,3]           => 2
[3,1,3,1,1]         => 1
[3,1,3,2]           => 1
[3,1,4,1]           => 1
[3,1,5]             => 1
[3,2,1,1,1,1]       => 0
[3,2,1,1,2]         => 1
[3,2,1,2,1]         => 1
[3,2,1,3]           => 1
[3,2,2,1,1]         => 0
[3,2,2,2]           => 0
[3,2,3,1]           => 1
[3,2,4]             => 1
[3,3,1,1,1]         => 0
[3,3,1,2]           => 1
[3,3,2,1]           => 0
[3,3,3]             => 0
[3,4,1,1]           => 1
[3,4,2]             => 1
[3,5,1]             => 1
[3,6]               => 1
[4,1,1,1,1,1]       => 0
[4,1,1,1,2]         => 1
[4,1,1,2,1]         => 1
[4,1,1,3]           => 1
[4,1,2,1,1]         => 1
[4,1,2,2]           => 1
[4,1,3,1]           => 1
[4,1,4]             => 1
[4,2,1,1,1]         => 0
[4,2,1,2]           => 1
[4,2,2,1]           => 0
[4,2,3]             => 1
[4,3,1,1]           => 0
[4,3,2]             => 0
[4,4,1]             => 0
[4,5]               => 1
[5,1,1,1,1]         => 0
[5,1,1,2]           => 1
[5,1,2,1]           => 1
[5,1,3]             => 1
[5,2,1,1]           => 0
[5,2,2]             => 0
[5,3,1]             => 0
[5,4]               => 0
[6,1,1,1]           => 0
[6,1,2]             => 1
[6,2,1]             => 0
[6,3]               => 0
[7,1,1]             => 0
[7,2]               => 0
[8,1]               => 0
[9]                 => 0

-----------------------------------------------------------------------------
Created: Apr 08, 2017 at 22:12 by Martin Rubey

-----------------------------------------------------------------------------
Last Updated: Nov 06, 2017 at 23:38 by Martin Rubey