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Identifier
Values
=>
[1]=>1 [1,1]=>1 [2]=>2 [1,1,1]=>1 [1,2]=>2 [2,1]=>2 [3]=>3 [1,1,1,1]=>1 [1,1,2]=>2 [1,2,1]=>2 [1,3]=>3 [2,1,1]=>2 [2,2]=>2 [3,1]=>3 [4]=>4 [1,1,1,1,1]=>1 [1,1,1,2]=>2 [1,1,2,1]=>2 [1,1,3]=>3 [1,2,1,1]=>2 [1,2,2]=>2 [1,3,1]=>3 [1,4]=>4 [2,1,1,1]=>2 [2,1,2]=>3 [2,2,1]=>2 [2,3]=>3 [3,1,1]=>3 [3,2]=>3 [4,1]=>4 [5]=>5 [1,1,1,1,1,1]=>1 [1,1,1,1,2]=>2 [1,1,1,2,1]=>2 [1,1,1,3]=>3 [1,1,2,1,1]=>2 [1,1,2,2]=>2 [1,1,3,1]=>3 [1,1,4]=>4 [1,2,1,1,1]=>2 [1,2,1,2]=>3 [1,2,2,1]=>2 [1,2,3]=>3 [1,3,1,1]=>3 [1,3,2]=>3 [1,4,1]=>4 [1,5]=>5 [2,1,1,1,1]=>2 [2,1,1,2]=>3 [2,1,2,1]=>3 [2,1,3]=>4 [2,2,1,1]=>2 [2,2,2]=>2 [2,3,1]=>3 [2,4]=>4 [3,1,1,1]=>3 [3,1,2]=>4 [3,2,1]=>3 [3,3]=>3 [4,1,1]=>4 [4,2]=>4 [5,1]=>5 [6]=>6 [1,1,1,1,1,1,1]=>1 [1,1,1,1,1,2]=>2 [1,1,1,1,2,1]=>2 [1,1,1,1,3]=>3 [1,1,1,2,1,1]=>2 [1,1,1,2,2]=>2 [1,1,1,3,1]=>3 [1,1,1,4]=>4 [1,1,2,1,1,1]=>2 [1,1,2,1,2]=>3 [1,1,2,2,1]=>2 [1,1,2,3]=>3 [1,1,3,1,1]=>3 [1,1,3,2]=>3 [1,1,4,1]=>4 [1,1,5]=>5 [1,2,1,1,1,1]=>2 [1,2,1,1,2]=>3 [1,2,1,2,1]=>3 [1,2,1,3]=>4 [1,2,2,1,1]=>2 [1,2,2,2]=>2 [1,2,3,1]=>3 [1,2,4]=>4 [1,3,1,1,1]=>3 [1,3,1,2]=>4 [1,3,2,1]=>3 [1,3,3]=>3 [1,4,1,1]=>4 [1,4,2]=>4 [1,5,1]=>5 [1,6]=>6 [2,1,1,1,1,1]=>2 [2,1,1,1,2]=>3 [2,1,1,2,1]=>3 [2,1,1,3]=>4 [2,1,2,1,1]=>3 [2,1,2,2]=>3 [2,1,3,1]=>4 [2,1,4]=>5 [2,2,1,1,1]=>2 [2,2,1,2]=>3 [2,2,2,1]=>2 [2,2,3]=>3 [2,3,1,1]=>3 [2,3,2]=>3 [2,4,1]=>4 [2,5]=>5 [3,1,1,1,1]=>3 [3,1,1,2]=>4 [3,1,2,1]=>4 [3,1,3]=>5 [3,2,1,1]=>3 [3,2,2]=>3 [3,3,1]=>3 [3,4]=>4 [4,1,1,1]=>4 [4,1,2]=>5 [4,2,1]=>4 [4,3]=>4 [5,1,1]=>5 [5,2]=>5 [6,1]=>6 [7]=>7 [1,1,1,1,1,1,1,1]=>1 [1,1,1,1,1,1,2]=>2 [1,1,1,1,1,2,1]=>2 [1,1,1,1,1,3]=>3 [1,1,1,1,2,1,1]=>2 [1,1,1,1,2,2]=>2 [1,1,1,1,3,1]=>3 [1,1,1,1,4]=>4 [1,1,1,2,1,1,1]=>2 [1,1,1,2,1,2]=>3 [1,1,1,2,2,1]=>2 [1,1,1,2,3]=>3 [1,1,1,3,1,1]=>3 [1,1,1,3,2]=>3 [1,1,1,4,1]=>4 [1,1,1,5]=>5 [1,1,2,1,1,1,1]=>2 [1,1,2,1,1,2]=>3 [1,1,2,1,2,1]=>3 [1,1,2,1,3]=>4 [1,1,2,2,1,1]=>2 [1,1,2,2,2]=>2 [1,1,2,3,1]=>3 [1,1,2,4]=>4 [1,1,3,1,1,1]=>3 [1,1,3,1,2]=>4 [1,1,3,2,1]=>3 [1,1,3,3]=>3 [1,1,4,1,1]=>4 [1,1,4,2]=>4 [1,1,5,1]=>5 [1,1,6]=>6 [1,2,1,1,1,1,1]=>2 [1,2,1,1,1,2]=>3 [1,2,1,1,2,1]=>3 [1,2,1,1,3]=>4 [1,2,1,2,1,1]=>3 [1,2,1,2,2]=>3 [1,2,1,3,1]=>4 [1,2,1,4]=>5 [1,2,2,1,1,1]=>2 [1,2,2,1,2]=>3 [1,2,2,2,1]=>2 [1,2,2,3]=>3 [1,2,3,1,1]=>3 [1,2,3,2]=>3 [1,2,4,1]=>4 [1,2,5]=>5 [1,3,1,1,1,1]=>3 [1,3,1,1,2]=>4 [1,3,1,2,1]=>4 [1,3,1,3]=>5 [1,3,2,1,1]=>3 [1,3,2,2]=>3 [1,3,3,1]=>3 [1,3,4]=>4 [1,4,1,1,1]=>4 [1,4,1,2]=>5 [1,4,2,1]=>4 [1,4,3]=>4 [1,5,1,1]=>5 [1,5,2]=>5 [1,6,1]=>6 [1,7]=>7 [2,1,1,1,1,1,1]=>2 [2,1,1,1,1,2]=>3 [2,1,1,1,2,1]=>3 [2,1,1,1,3]=>4 [2,1,1,2,1,1]=>3 [2,1,1,2,2]=>3 [2,1,1,3,1]=>4 [2,1,1,4]=>5 [2,1,2,1,1,1]=>3 [2,1,2,1,2]=>4 [2,1,2,2,1]=>3 [2,1,2,3]=>4 [2,1,3,1,1]=>4 [2,1,3,2]=>4 [2,1,4,1]=>5 [2,1,5]=>6 [2,2,1,1,1,1]=>2 [2,2,1,1,2]=>3 [2,2,1,2,1]=>3 [2,2,1,3]=>4 [2,2,2,1,1]=>2 [2,2,2,2]=>2 [2,2,3,1]=>3 [2,2,4]=>4 [2,3,1,1,1]=>3 [2,3,1,2]=>4 [2,3,2,1]=>3 [2,3,3]=>3 [2,4,1,1]=>4 [2,4,2]=>4 [2,5,1]=>5 [2,6]=>6 [3,1,1,1,1,1]=>3 [3,1,1,1,2]=>4 [3,1,1,2,1]=>4 [3,1,1,3]=>5 [3,1,2,1,1]=>4 [3,1,2,2]=>4 [3,1,3,1]=>5 [3,1,4]=>6 [3,2,1,1,1]=>3 [3,2,1,2]=>4 [3,2,2,1]=>3 [3,2,3]=>4 [3,3,1,1]=>3 [3,3,2]=>3 [3,4,1]=>4 [3,5]=>5 [4,1,1,1,1]=>4 [4,1,1,2]=>5 [4,1,2,1]=>5 [4,1,3]=>6 [4,2,1,1]=>4 [4,2,2]=>4 [4,3,1]=>4 [4,4]=>4 [5,1,1,1]=>5 [5,1,2]=>6 [5,2,1]=>5 [5,3]=>5 [6,1,1]=>6 [6,2]=>6 [7,1]=>7 [8]=>8 [1,1,1,1,1,1,1,1,1]=>1 [1,1,1,1,1,1,1,2]=>2 [1,1,1,1,1,1,2,1]=>2 [1,1,1,1,1,1,3]=>3 [1,1,1,1,1,2,1,1]=>2 [1,1,1,1,1,2,2]=>2 [1,1,1,1,1,3,1]=>3 [1,1,1,1,1,4]=>4 [1,1,1,1,2,1,1,1]=>2 [1,1,1,1,2,1,2]=>3 [1,1,1,1,2,2,1]=>2 [1,1,1,1,2,3]=>3 [1,1,1,1,3,1,1]=>3 [1,1,1,1,3,2]=>3 [1,1,1,1,4,1]=>4 [1,1,1,1,5]=>5 [1,1,1,2,1,1,1,1]=>2 [1,1,1,2,1,1,2]=>3 [1,1,1,2,1,2,1]=>3 [1,1,1,2,1,3]=>4 [1,1,1,2,2,1,1]=>2 [1,1,1,2,2,2]=>2 [1,1,1,2,3,1]=>3 [1,1,1,2,4]=>4 [1,1,1,3,1,1,1]=>3 [1,1,1,3,1,2]=>4 [1,1,1,3,2,1]=>3 [1,1,1,3,3]=>3 [1,1,1,4,1,1]=>4 [1,1,1,4,2]=>4 [1,1,1,5,1]=>5 [1,1,1,6]=>6 [1,1,2,1,1,1,1,1]=>2 [1,1,2,1,1,1,2]=>3 [1,1,2,1,1,2,1]=>3 [1,1,2,1,1,3]=>4 [1,1,2,1,2,1,1]=>3 [1,1,2,1,2,2]=>3 [1,1,2,1,3,1]=>4 [1,1,2,1,4]=>5 [1,1,2,2,1,1,1]=>2 [1,1,2,2,1,2]=>3 [1,1,2,2,2,1]=>2 [1,1,2,2,3]=>3 [1,1,2,3,1,1]=>3 [1,1,2,3,2]=>3 [1,1,2,4,1]=>4 [1,1,2,5]=>5 [1,1,3,1,1,1,1]=>3 [1,1,3,1,1,2]=>4 [1,1,3,1,2,1]=>4 [1,1,3,1,3]=>5 [1,1,3,2,1,1]=>3 [1,1,3,2,2]=>3 [1,1,3,3,1]=>3 [1,1,3,4]=>4 [1,1,4,1,1,1]=>4 [1,1,4,1,2]=>5 [1,1,4,2,1]=>4 [1,1,4,3]=>4 [1,1,5,1,1]=>5 [1,1,5,2]=>5 [1,1,6,1]=>6 [1,1,7]=>7 [1,2,1,1,1,1,1,1]=>2 [1,2,1,1,1,1,2]=>3 [1,2,1,1,1,2,1]=>3 [1,2,1,1,1,3]=>4 [1,2,1,1,2,1,1]=>3 [1,2,1,1,2,2]=>3 [1,2,1,1,3,1]=>4 [1,2,1,1,4]=>5 [1,2,1,2,1,1,1]=>3 [1,2,1,2,1,2]=>4 [1,2,1,2,2,1]=>3 [1,2,1,2,3]=>4 [1,2,1,3,1,1]=>4 [1,2,1,3,2]=>4 [1,2,1,4,1]=>5 [1,2,1,5]=>6 [1,2,2,1,1,1,1]=>2 [1,2,2,1,1,2]=>3 [1,2,2,1,2,1]=>3 [1,2,2,1,3]=>4 [1,2,2,2,1,1]=>2 [1,2,2,2,2]=>2 [1,2,2,3,1]=>3 [1,2,2,4]=>4 [1,2,3,1,1,1]=>3 [1,2,3,1,2]=>4 [1,2,3,2,1]=>3 [1,2,3,3]=>3 [1,2,4,1,1]=>4 [1,2,4,2]=>4 [1,2,5,1]=>5 [1,2,6]=>6 [1,3,1,1,1,1,1]=>3 [1,3,1,1,1,2]=>4 [1,3,1,1,2,1]=>4 [1,3,1,1,3]=>5 [1,3,1,2,1,1]=>4 [1,3,1,2,2]=>4 [1,3,1,3,1]=>5 [1,3,1,4]=>6 [1,3,2,1,1,1]=>3 [1,3,2,1,2]=>4 [1,3,2,2,1]=>3 [1,3,2,3]=>4 [1,3,3,1,1]=>3 [1,3,3,2]=>3 [1,3,4,1]=>4 [1,3,5]=>5 [1,4,1,1,1,1]=>4 [1,4,1,1,2]=>5 [1,4,1,2,1]=>5 [1,4,1,3]=>6 [1,4,2,1,1]=>4 [1,4,2,2]=>4 [1,4,3,1]=>4 [1,4,4]=>4 [1,5,1,1,1]=>5 [1,5,1,2]=>6 [1,5,2,1]=>5 [1,5,3]=>5 [1,6,1,1]=>6 [1,6,2]=>6 [1,7,1]=>7 [1,8]=>8 [2,1,1,1,1,1,1,1]=>2 [2,1,1,1,1,1,2]=>3 [2,1,1,1,1,2,1]=>3 [2,1,1,1,1,3]=>4 [2,1,1,1,2,1,1]=>3 [2,1,1,1,2,2]=>3 [2,1,1,1,3,1]=>4 [2,1,1,1,4]=>5 [2,1,1,2,1,1,1]=>3 [2,1,1,2,1,2]=>4 [2,1,1,2,2,1]=>3 [2,1,1,2,3]=>4 [2,1,1,3,1,1]=>4 [2,1,1,3,2]=>4 [2,1,1,4,1]=>5 [2,1,1,5]=>6 [2,1,2,1,1,1,1]=>3 [2,1,2,1,1,2]=>4 [2,1,2,1,2,1]=>4 [2,1,2,1,3]=>5 [2,1,2,2,1,1]=>3 [2,1,2,2,2]=>3 [2,1,2,3,1]=>4 [2,1,2,4]=>5 [2,1,3,1,1,1]=>4 [2,1,3,1,2]=>5 [2,1,3,2,1]=>4 [2,1,3,3]=>4 [2,1,4,1,1]=>5 [2,1,4,2]=>5 [2,1,5,1]=>6 [2,1,6]=>7 [2,2,1,1,1,1,1]=>2 [2,2,1,1,1,2]=>3 [2,2,1,1,2,1]=>3 [2,2,1,1,3]=>4 [2,2,1,2,1,1]=>3 [2,2,1,2,2]=>3 [2,2,1,3,1]=>4 [2,2,1,4]=>5 [2,2,2,1,1,1]=>2 [2,2,2,1,2]=>3 [2,2,2,2,1]=>2 [2,2,2,3]=>3 [2,2,3,1,1]=>3 [2,2,3,2]=>3 [2,2,4,1]=>4 [2,2,5]=>5 [2,3,1,1,1,1]=>3 [2,3,1,1,2]=>4 [2,3,1,2,1]=>4 [2,3,1,3]=>5 [2,3,2,1,1]=>3 [2,3,2,2]=>3 [2,3,3,1]=>3 [2,3,4]=>4 [2,4,1,1,1]=>4 [2,4,1,2]=>5 [2,4,2,1]=>4 [2,4,3]=>4 [2,5,1,1]=>5 [2,5,2]=>5 [2,6,1]=>6 [2,7]=>7 [3,1,1,1,1,1,1]=>3 [3,1,1,1,1,2]=>4 [3,1,1,1,2,1]=>4 [3,1,1,1,3]=>5 [3,1,1,2,1,1]=>4 [3,1,1,2,2]=>4 [3,1,1,3,1]=>5 [3,1,1,4]=>6 [3,1,2,1,1,1]=>4 [3,1,2,1,2]=>5 [3,1,2,2,1]=>4 [3,1,2,3]=>5 [3,1,3,1,1]=>5 [3,1,3,2]=>5 [3,1,4,1]=>6 [3,1,5]=>7 [3,2,1,1,1,1]=>3 [3,2,1,1,2]=>4 [3,2,1,2,1]=>4 [3,2,1,3]=>5 [3,2,2,1,1]=>3 [3,2,2,2]=>3 [3,2,3,1]=>4 [3,2,4]=>5 [3,3,1,1,1]=>3 [3,3,1,2]=>4 [3,3,2,1]=>3 [3,3,3]=>3 [3,4,1,1]=>4 [3,4,2]=>4 [3,5,1]=>5 [3,6]=>6 [4,1,1,1,1,1]=>4 [4,1,1,1,2]=>5 [4,1,1,2,1]=>5 [4,1,1,3]=>6 [4,1,2,1,1]=>5 [4,1,2,2]=>5 [4,1,3,1]=>6 [4,1,4]=>7 [4,2,1,1,1]=>4 [4,2,1,2]=>5 [4,2,2,1]=>4 [4,2,3]=>5 [4,3,1,1]=>4 [4,3,2]=>4 [4,4,1]=>4 [4,5]=>5 [5,1,1,1,1]=>5 [5,1,1,2]=>6 [5,1,2,1]=>6 [5,1,3]=>7 [5,2,1,1]=>5 [5,2,2]=>5 [5,3,1]=>5 [5,4]=>5 [6,1,1,1]=>6 [6,1,2]=>7 [6,2,1]=>6 [6,3]=>6 [7,1,1]=>7 [7,2]=>7 [8,1]=>8 [9]=>9
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Description
The number of up steps of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.
References
[1] Deutsch, E., Elizalde, S. A bijection between bargraphs and Dyck paths arXiv:1705.05984
Code
def statistic(c):
    if len(c) == 0:
        return 0
    U = c[0]
    for i in range(1, len(c)):
        if c[i] > c[i-1]:
            U += c[i]-c[i-1]
    return U

Created
May 18, 2017 at 09:45 by Martin Rubey
Updated
Nov 19, 2017 at 22:04 by Christian Stump