Identifier
Identifier
Values
[1] => 1
[1,1] => 1
[2] => 1
[1,1,1] => 1
[1,2] => 2
[2,1] => 1
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 2
[1,2,1] => 2
[1,3] => 2
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 2
[1,1,2,1] => 2
[1,1,3] => 2
[1,2,1,1] => 2
[1,2,2] => 2
[1,3,1] => 2
[1,4] => 2
[2,1,1,1] => 1
[2,1,2] => 2
[2,2,1] => 2
[2,3] => 2
[3,1,1] => 1
[3,2] => 2
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 2
[1,1,1,2,1] => 2
[1,1,1,3] => 2
[1,1,2,1,1] => 2
[1,1,2,2] => 2
[1,1,3,1] => 2
[1,1,4] => 2
[1,2,1,1,1] => 2
[1,2,1,2] => 2
[1,2,2,1] => 2
[1,2,3] => 3
[1,3,1,1] => 2
[1,3,2] => 2
[1,4,1] => 2
[1,5] => 2
[2,1,1,1,1] => 1
[2,1,1,2] => 2
[2,1,2,1] => 2
[2,1,3] => 2
[2,2,1,1] => 2
[2,2,2] => 2
[2,3,1] => 2
[2,4] => 2
[3,1,1,1] => 1
[3,1,2] => 2
[3,2,1] => 2
[3,3] => 2
[4,1,1] => 1
[4,2] => 2
[5,1] => 1
[6] => 1
[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] => 2
[1,1,1,2,1,1] => 2
[1,1,1,2,2] => 2
[1,1,1,3,1] => 2
[1,1,1,4] => 2
[1,1,2,1,1,1] => 2
[1,1,2,1,2] => 2
[1,1,2,2,1] => 2
[1,1,2,3] => 3
[1,1,3,1,1] => 2
[1,1,3,2] => 2
[1,1,4,1] => 2
[1,1,5] => 2
[1,2,1,1,1,1] => 2
[1,2,1,1,2] => 2
[1,2,1,2,1] => 2
[1,2,1,3] => 3
[1,2,2,1,1] => 2
[1,2,2,2] => 2
[1,2,3,1] => 3
[1,2,4] => 3
[1,3,1,1,1] => 2
[1,3,1,2] => 2
[1,3,2,1] => 2
[1,3,3] => 3
[1,4,1,1] => 2
[1,4,2] => 2
[1,5,1] => 2
[1,6] => 2
[2,1,1,1,1,1] => 1
[2,1,1,1,2] => 2
[2,1,1,2,1] => 2
[2,1,1,3] => 2
[2,1,2,1,1] => 2
[2,1,2,2] => 2
[2,1,3,1] => 2
[2,1,4] => 2
[2,2,1,1,1] => 2
[2,2,1,2] => 2
[2,2,2,1] => 2
[2,2,3] => 3
[2,3,1,1] => 2
[2,3,2] => 2
[2,4,1] => 2
[2,5] => 2
[3,1,1,1,1] => 1
[3,1,1,2] => 2
[3,1,2,1] => 2
[3,1,3] => 2
[3,2,1,1] => 2
[3,2,2] => 2
[3,3,1] => 2
[3,4] => 2
[4,1,1,1] => 1
[4,1,2] => 2
[4,2,1] => 2
[4,3] => 2
[5,1,1] => 1
[5,2] => 2
[6,1] => 1
[7] => 1
[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] => 2
[1,1,1,1,2,1,1] => 2
[1,1,1,1,2,2] => 2
[1,1,1,1,3,1] => 2
[1,1,1,1,4] => 2
[1,1,1,2,1,1,1] => 2
[1,1,1,2,1,2] => 2
[1,1,1,2,2,1] => 2
[1,1,1,2,3] => 3
[1,1,1,3,1,1] => 2
[1,1,1,3,2] => 2
[1,1,1,4,1] => 2
[1,1,1,5] => 2
[1,1,2,1,1,1,1] => 2
[1,1,2,1,1,2] => 2
[1,1,2,1,2,1] => 2
[1,1,2,1,3] => 3
[1,1,2,2,1,1] => 2
[1,1,2,2,2] => 2
[1,1,2,3,1] => 3
[1,1,2,4] => 3
[1,1,3,1,1,1] => 2
[1,1,3,1,2] => 2
[1,1,3,2,1] => 2
[1,1,3,3] => 3
[1,1,4,1,1] => 2
[1,1,4,2] => 2
[1,1,5,1] => 2
[1,1,6] => 2
[1,2,1,1,1,1,1] => 2
[1,2,1,1,1,2] => 2
[1,2,1,1,2,1] => 2
[1,2,1,1,3] => 3
[1,2,1,2,1,1] => 2
[1,2,1,2,2] => 2
[1,2,1,3,1] => 3
[1,2,1,4] => 3
[1,2,2,1,1,1] => 2
[1,2,2,1,2] => 2
[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] => 3
[1,2,5] => 3
[1,3,1,1,1,1] => 2
[1,3,1,1,2] => 2
[1,3,1,2,1] => 2
[1,3,1,3] => 3
[1,3,2,1,1] => 2
[1,3,2,2] => 2
[1,3,3,1] => 3
[1,3,4] => 3
[1,4,1,1,1] => 2
[1,4,1,2] => 2
[1,4,2,1] => 2
[1,4,3] => 3
[1,5,1,1] => 2
[1,5,2] => 2
[1,6,1] => 2
[1,7] => 2
[2,1,1,1,1,1,1] => 1
[2,1,1,1,1,2] => 2
[2,1,1,1,2,1] => 2
[2,1,1,1,3] => 2
[2,1,1,2,1,1] => 2
[2,1,1,2,2] => 2
[2,1,1,3,1] => 2
[2,1,1,4] => 2
[2,1,2,1,1,1] => 2
[2,1,2,1,2] => 2
[2,1,2,2,1] => 2
[2,1,2,3] => 3
[2,1,3,1,1] => 2
[2,1,3,2] => 2
[2,1,4,1] => 2
[2,1,5] => 2
[2,2,1,1,1,1] => 2
[2,2,1,1,2] => 2
[2,2,1,2,1] => 2
[2,2,1,3] => 3
[2,2,2,1,1] => 2
[2,2,2,2] => 2
[2,2,3,1] => 3
[2,2,4] => 3
[2,3,1,1,1] => 2
[2,3,1,2] => 2
[2,3,2,1] => 2
[2,3,3] => 3
[2,4,1,1] => 2
[2,4,2] => 2
[2,5,1] => 2
[2,6] => 2
[3,1,1,1,1,1] => 1
[3,1,1,1,2] => 2
[3,1,1,2,1] => 2
[3,1,1,3] => 2
[3,1,2,1,1] => 2
[3,1,2,2] => 2
[3,1,3,1] => 2
[3,1,4] => 2
[3,2,1,1,1] => 2
[3,2,1,2] => 2
[3,2,2,1] => 2
[3,2,3] => 3
[3,3,1,1] => 2
[3,3,2] => 2
[3,4,1] => 2
[3,5] => 2
[4,1,1,1,1] => 1
[4,1,1,2] => 2
[4,1,2,1] => 2
[4,1,3] => 2
[4,2,1,1] => 2
[4,2,2] => 2
[4,3,1] => 2
[4,4] => 2
[5,1,1,1] => 1
[5,1,2] => 2
[5,2,1] => 2
[5,3] => 2
[6,1,1] => 1
[6,2] => 2
[7,1] => 1
[8] => 1
[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] => 2
[1,1,1,1,1,2,1,1] => 2
[1,1,1,1,1,2,2] => 2
[1,1,1,1,1,3,1] => 2
[1,1,1,1,1,4] => 2
[1,1,1,1,2,1,1,1] => 2
[1,1,1,1,2,1,2] => 2
[1,1,1,1,2,2,1] => 2
[1,1,1,1,2,3] => 3
[1,1,1,1,3,1,1] => 2
[1,1,1,1,3,2] => 2
[1,1,1,1,4,1] => 2
[1,1,1,1,5] => 2
[1,1,1,2,1,1,1,1] => 2
[1,1,1,2,1,1,2] => 2
[1,1,1,2,1,2,1] => 2
[1,1,1,2,1,3] => 3
[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] => 3
[1,1,1,3,1,1,1] => 2
[1,1,1,3,1,2] => 2
[1,1,1,3,2,1] => 2
[1,1,1,3,3] => 3
[1,1,1,4,1,1] => 2
[1,1,1,4,2] => 2
[1,1,1,5,1] => 2
[1,1,1,6] => 2
[1,1,2,1,1,1,1,1] => 2
[1,1,2,1,1,1,2] => 2
[1,1,2,1,1,2,1] => 2
[1,1,2,1,1,3] => 3
[1,1,2,1,2,1,1] => 2
[1,1,2,1,2,2] => 2
[1,1,2,1,3,1] => 3
[1,1,2,1,4] => 3
[1,1,2,2,1,1,1] => 2
[1,1,2,2,1,2] => 2
[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] => 3
[1,1,2,5] => 3
[1,1,3,1,1,1,1] => 2
[1,1,3,1,1,2] => 2
[1,1,3,1,2,1] => 2
[1,1,3,1,3] => 3
[1,1,3,2,1,1] => 2
[1,1,3,2,2] => 2
[1,1,3,3,1] => 3
[1,1,3,4] => 3
[1,1,4,1,1,1] => 2
[1,1,4,1,2] => 2
[1,1,4,2,1] => 2
[1,1,4,3] => 3
[1,1,5,1,1] => 2
[1,1,5,2] => 2
[1,1,6,1] => 2
[1,1,7] => 2
[1,2,1,1,1,1,1,1] => 2
[1,2,1,1,1,1,2] => 2
[1,2,1,1,1,2,1] => 2
[1,2,1,1,1,3] => 3
[1,2,1,1,2,1,1] => 2
[1,2,1,1,2,2] => 2
[1,2,1,1,3,1] => 3
[1,2,1,1,4] => 3
[1,2,1,2,1,1,1] => 2
[1,2,1,2,1,2] => 2
[1,2,1,2,2,1] => 2
[1,2,1,2,3] => 3
[1,2,1,3,1,1] => 3
[1,2,1,3,2] => 3
[1,2,1,4,1] => 3
[1,2,1,5] => 3
[1,2,2,1,1,1,1] => 2
[1,2,2,1,1,2] => 2
[1,2,2,1,2,1] => 2
[1,2,2,1,3] => 3
[1,2,2,2,1,1] => 2
[1,2,2,2,2] => 2
[1,2,2,3,1] => 3
[1,2,2,4] => 3
[1,2,3,1,1,1] => 3
[1,2,3,1,2] => 3
[1,2,3,2,1] => 3
[1,2,3,3] => 3
[1,2,4,1,1] => 3
[1,2,4,2] => 3
[1,2,5,1] => 3
[1,2,6] => 3
[1,3,1,1,1,1,1] => 2
[1,3,1,1,1,2] => 2
[1,3,1,1,2,1] => 2
[1,3,1,1,3] => 3
[1,3,1,2,1,1] => 2
[1,3,1,2,2] => 2
[1,3,1,3,1] => 3
[1,3,1,4] => 3
[1,3,2,1,1,1] => 2
[1,3,2,1,2] => 2
[1,3,2,2,1] => 2
[1,3,2,3] => 3
[1,3,3,1,1] => 3
[1,3,3,2] => 3
[1,3,4,1] => 3
[1,3,5] => 3
[1,4,1,1,1,1] => 2
[1,4,1,1,2] => 2
[1,4,1,2,1] => 2
[1,4,1,3] => 3
[1,4,2,1,1] => 2
[1,4,2,2] => 2
[1,4,3,1] => 3
[1,4,4] => 3
[1,5,1,1,1] => 2
[1,5,1,2] => 2
[1,5,2,1] => 2
[1,5,3] => 3
[1,6,1,1] => 2
[1,6,2] => 2
[1,7,1] => 2
[1,8] => 2
[2,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,2] => 2
[2,1,1,1,1,2,1] => 2
[2,1,1,1,1,3] => 2
[2,1,1,1,2,1,1] => 2
[2,1,1,1,2,2] => 2
[2,1,1,1,3,1] => 2
[2,1,1,1,4] => 2
[2,1,1,2,1,1,1] => 2
[2,1,1,2,1,2] => 2
[2,1,1,2,2,1] => 2
[2,1,1,2,3] => 3
[2,1,1,3,1,1] => 2
[2,1,1,3,2] => 2
[2,1,1,4,1] => 2
[2,1,1,5] => 2
[2,1,2,1,1,1,1] => 2
[2,1,2,1,1,2] => 2
[2,1,2,1,2,1] => 2
[2,1,2,1,3] => 3
[2,1,2,2,1,1] => 2
[2,1,2,2,2] => 2
[2,1,2,3,1] => 3
[2,1,2,4] => 3
[2,1,3,1,1,1] => 2
[2,1,3,1,2] => 2
[2,1,3,2,1] => 2
[2,1,3,3] => 3
[2,1,4,1,1] => 2
[2,1,4,2] => 2
[2,1,5,1] => 2
[2,1,6] => 2
[2,2,1,1,1,1,1] => 2
[2,2,1,1,1,2] => 2
[2,2,1,1,2,1] => 2
[2,2,1,1,3] => 3
[2,2,1,2,1,1] => 2
[2,2,1,2,2] => 2
[2,2,1,3,1] => 3
[2,2,1,4] => 3
[2,2,2,1,1,1] => 2
[2,2,2,1,2] => 2
[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] => 3
[2,2,5] => 3
[2,3,1,1,1,1] => 2
[2,3,1,1,2] => 2
[2,3,1,2,1] => 2
[2,3,1,3] => 3
[2,3,2,1,1] => 2
[2,3,2,2] => 2
[2,3,3,1] => 3
[2,3,4] => 3
[2,4,1,1,1] => 2
[2,4,1,2] => 2
[2,4,2,1] => 2
[2,4,3] => 3
[2,5,1,1] => 2
[2,5,2] => 2
[2,6,1] => 2
[2,7] => 2
[3,1,1,1,1,1,1] => 1
[3,1,1,1,1,2] => 2
[3,1,1,1,2,1] => 2
[3,1,1,1,3] => 2
[3,1,1,2,1,1] => 2
[3,1,1,2,2] => 2
[3,1,1,3,1] => 2
[3,1,1,4] => 2
[3,1,2,1,1,1] => 2
[3,1,2,1,2] => 2
[3,1,2,2,1] => 2
[3,1,2,3] => 3
[3,1,3,1,1] => 2
[3,1,3,2] => 2
[3,1,4,1] => 2
[3,1,5] => 2
[3,2,1,1,1,1] => 2
[3,2,1,1,2] => 2
[3,2,1,2,1] => 2
[3,2,1,3] => 3
[3,2,2,1,1] => 2
[3,2,2,2] => 2
[3,2,3,1] => 3
[3,2,4] => 3
[3,3,1,1,1] => 2
[3,3,1,2] => 2
[3,3,2,1] => 2
[3,3,3] => 3
[3,4,1,1] => 2
[3,4,2] => 2
[3,5,1] => 2
[3,6] => 2
[4,1,1,1,1,1] => 1
[4,1,1,1,2] => 2
[4,1,1,2,1] => 2
[4,1,1,3] => 2
[4,1,2,1,1] => 2
[4,1,2,2] => 2
[4,1,3,1] => 2
[4,1,4] => 2
[4,2,1,1,1] => 2
[4,2,1,2] => 2
[4,2,2,1] => 2
[4,2,3] => 3
[4,3,1,1] => 2
[4,3,2] => 2
[4,4,1] => 2
[4,5] => 2
[5,1,1,1,1] => 1
[5,1,1,2] => 2
[5,1,2,1] => 2
[5,1,3] => 2
[5,2,1,1] => 2
[5,2,2] => 2
[5,3,1] => 2
[5,4] => 2
[6,1,1,1] => 1
[6,1,2] => 2
[6,2,1] => 2
[6,3] => 2
[7,1,1] => 1
[7,2] => 2
[8,1] => 1
[9] => 1
click to show generating function       
Description
The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
References
[1] Skandera, M. An Eulerian partner for inversions MathSciNet:1848722
Code
def statistic(w):
    l = 0
    for e in w:
        if e > l:
            l += 1
    return l

Created
Apr 09, 2017 at 10:46 by Martin Rubey
Updated
Nov 06, 2017 at 23:33 by Martin Rubey