edit this statistic or download as text // json
Identifier
Values
=>
Cc0002;cc-rep
[]=>1 [1]=>2 [2]=>1 [1,1]=>2 [3]=>1 [2,1]=>3 [1,1,1]=>2 [4]=>1 [3,1]=>2 [2,2]=>1 [2,1,1]=>3 [1,1,1,1]=>2 [5]=>1 [4,1]=>2 [3,2]=>1 [3,1,1]=>2 [2,2,1]=>3 [2,1,1,1]=>3 [1,1,1,1,1]=>2 [6]=>1 [5,1]=>2 [4,2]=>1 [4,1,1]=>2 [3,3]=>1 [3,2,1]=>4 [3,1,1,1]=>2 [2,2,2]=>1 [2,2,1,1]=>3 [2,1,1,1,1]=>3 [1,1,1,1,1,1]=>2 [7]=>1 [6,1]=>2 [5,2]=>1 [5,1,1]=>2 [4,3]=>1 [4,2,1]=>3 [4,1,1,1]=>2 [3,3,1]=>2 [3,2,2]=>1 [3,2,1,1]=>4 [3,1,1,1,1]=>2 [2,2,2,1]=>3 [2,2,1,1,1]=>3 [2,1,1,1,1,1]=>3 [1,1,1,1,1,1,1]=>2 [8]=>1 [7,1]=>2 [6,2]=>1 [6,1,1]=>2 [5,3]=>1 [5,2,1]=>3 [5,1,1,1]=>2 [4,4]=>1 [4,3,1]=>2 [4,2,2]=>1 [4,2,1,1]=>3 [4,1,1,1,1]=>2 [3,3,2]=>1 [3,3,1,1]=>2 [3,2,2,1]=>4 [3,2,1,1,1]=>4 [3,1,1,1,1,1]=>2 [2,2,2,2]=>1 [2,2,2,1,1]=>3 [2,2,1,1,1,1]=>3 [2,1,1,1,1,1,1]=>3 [1,1,1,1,1,1,1,1]=>2 [9]=>1 [8,1]=>2 [7,2]=>1 [7,1,1]=>2 [6,3]=>1 [6,2,1]=>3 [6,1,1,1]=>2 [5,4]=>1 [5,3,1]=>2 [5,2,2]=>1 [5,2,1,1]=>3 [5,1,1,1,1]=>2 [4,4,1]=>2 [4,3,2]=>1 [4,3,1,1]=>2 [4,2,2,1]=>3 [4,2,1,1,1]=>3 [4,1,1,1,1,1]=>2 [3,3,3]=>1 [3,3,2,1]=>4 [3,3,1,1,1]=>2 [3,2,2,2]=>1 [3,2,2,1,1]=>4 [3,2,1,1,1,1]=>4 [3,1,1,1,1,1,1]=>2 [2,2,2,2,1]=>3 [2,2,2,1,1,1]=>3 [2,2,1,1,1,1,1]=>3 [2,1,1,1,1,1,1,1]=>3 [1,1,1,1,1,1,1,1,1]=>2 [10]=>1 [9,1]=>2 [8,2]=>1 [8,1,1]=>2 [7,3]=>1 [7,2,1]=>3 [7,1,1,1]=>2 [6,4]=>1 [6,3,1]=>2 [6,2,2]=>1 [6,2,1,1]=>3 [6,1,1,1,1]=>2 [5,5]=>1 [5,4,1]=>2 [5,3,2]=>1 [5,3,1,1]=>2 [5,2,2,1]=>3 [5,2,1,1,1]=>3 [5,1,1,1,1,1]=>2 [4,4,2]=>1 [4,4,1,1]=>2 [4,3,3]=>1 [4,3,2,1]=>5 [4,3,1,1,1]=>2 [4,2,2,2]=>1 [4,2,2,1,1]=>3 [4,2,1,1,1,1]=>3 [4,1,1,1,1,1,1]=>2 [3,3,3,1]=>2 [3,3,2,2]=>1 [3,3,2,1,1]=>4 [3,3,1,1,1,1]=>2 [3,2,2,2,1]=>4 [3,2,2,1,1,1]=>4 [3,2,1,1,1,1,1]=>4 [3,1,1,1,1,1,1,1]=>2 [2,2,2,2,2]=>1 [2,2,2,2,1,1]=>3 [2,2,2,1,1,1,1]=>3 [2,2,1,1,1,1,1,1]=>3 [2,1,1,1,1,1,1,1,1]=>3 [1,1,1,1,1,1,1,1,1,1]=>2 [11]=>1 [10,1]=>2 [9,2]=>1 [9,1,1]=>2 [8,3]=>1 [8,2,1]=>3 [8,1,1,1]=>2 [7,4]=>1 [7,3,1]=>2 [7,2,2]=>1 [7,2,1,1]=>3 [7,1,1,1,1]=>2 [6,5]=>1 [6,4,1]=>2 [6,3,2]=>1 [6,3,1,1]=>2 [6,2,2,1]=>3 [6,2,1,1,1]=>3 [6,1,1,1,1,1]=>2 [5,5,1]=>2 [5,4,2]=>1 [5,4,1,1]=>2 [5,3,3]=>1 [5,3,2,1]=>4 [5,3,1,1,1]=>2 [5,2,2,2]=>1 [5,2,2,1,1]=>3 [5,2,1,1,1,1]=>3 [5,1,1,1,1,1,1]=>2 [4,4,3]=>1 [4,4,2,1]=>3 [4,4,1,1,1]=>2 [4,3,3,1]=>2 [4,3,2,2]=>1 [4,3,2,1,1]=>5 [4,3,1,1,1,1]=>2 [4,2,2,2,1]=>3 [4,2,2,1,1,1]=>3 [4,2,1,1,1,1,1]=>3 [4,1,1,1,1,1,1,1]=>2 [3,3,3,2]=>1 [3,3,3,1,1]=>2 [3,3,2,2,1]=>4 [3,3,2,1,1,1]=>4 [3,3,1,1,1,1,1]=>2 [3,2,2,2,2]=>1 [3,2,2,2,1,1]=>4 [3,2,2,1,1,1,1]=>4 [3,2,1,1,1,1,1,1]=>4 [3,1,1,1,1,1,1,1,1]=>2 [2,2,2,2,2,1]=>3 [2,2,2,2,1,1,1]=>3 [2,2,2,1,1,1,1,1]=>3 [2,2,1,1,1,1,1,1,1]=>3 [2,1,1,1,1,1,1,1,1,1]=>3 [1,1,1,1,1,1,1,1,1,1,1]=>2 [12]=>1 [11,1]=>2 [10,2]=>1 [10,1,1]=>2 [9,3]=>1 [9,2,1]=>3 [9,1,1,1]=>2 [8,4]=>1 [8,3,1]=>2 [8,2,2]=>1 [8,2,1,1]=>3 [8,1,1,1,1]=>2 [7,5]=>1 [7,4,1]=>2 [7,3,2]=>1 [7,3,1,1]=>2 [7,2,2,1]=>3 [7,2,1,1,1]=>3 [7,1,1,1,1,1]=>2 [6,6]=>1 [6,5,1]=>2 [6,4,2]=>1 [6,4,1,1]=>2 [6,3,3]=>1 [6,3,2,1]=>4 [6,3,1,1,1]=>2 [6,2,2,2]=>1 [6,2,2,1,1]=>3 [6,2,1,1,1,1]=>3 [6,1,1,1,1,1,1]=>2 [5,5,2]=>1 [5,5,1,1]=>2 [5,4,3]=>1 [5,4,2,1]=>3 [5,4,1,1,1]=>2 [5,3,3,1]=>2 [5,3,2,2]=>1 [5,3,2,1,1]=>4 [5,3,1,1,1,1]=>2 [5,2,2,2,1]=>3 [5,2,2,1,1,1]=>3 [5,2,1,1,1,1,1]=>3 [5,1,1,1,1,1,1,1]=>2 [4,4,4]=>1 [4,4,3,1]=>2 [4,4,2,2]=>1 [4,4,2,1,1]=>3 [4,4,1,1,1,1]=>2 [4,3,3,2]=>1 [4,3,3,1,1]=>2 [4,3,2,2,1]=>5 [4,3,2,1,1,1]=>5 [4,3,1,1,1,1,1]=>2 [4,2,2,2,2]=>1 [4,2,2,2,1,1]=>3 [4,2,2,1,1,1,1]=>3 [4,2,1,1,1,1,1,1]=>3 [4,1,1,1,1,1,1,1,1]=>2 [3,3,3,3]=>1 [3,3,3,2,1]=>4 [3,3,3,1,1,1]=>2 [3,3,2,2,2]=>1 [3,3,2,2,1,1]=>4 [3,3,2,1,1,1,1]=>4 [3,3,1,1,1,1,1,1]=>2 [3,2,2,2,2,1]=>4 [3,2,2,2,1,1,1]=>4 [3,2,2,1,1,1,1,1]=>4 [3,2,1,1,1,1,1,1,1]=>4 [3,1,1,1,1,1,1,1,1,1]=>2 [2,2,2,2,2,2]=>1 [2,2,2,2,2,1,1]=>3 [2,2,2,2,1,1,1,1]=>3 [2,2,2,1,1,1,1,1,1]=>3 [2,2,1,1,1,1,1,1,1,1]=>3 [2,1,1,1,1,1,1,1,1,1,1]=>3 [1,1,1,1,1,1,1,1,1,1,1,1]=>2 [8,5]=>1 [7,5,1]=>2 [7,4,2]=>1 [5,5,3]=>1 [5,4,4]=>1 [5,4,3,1]=>2 [5,4,2,2]=>1 [5,4,2,1,1]=>3 [5,4,1,1,1,1]=>2 [5,3,3,2]=>1 [5,3,3,1,1]=>2 [5,3,2,2,1]=>4 [5,3,2,1,1,1]=>4 [4,4,4,1]=>2 [4,4,3,2]=>1 [4,4,3,1,1]=>2 [4,4,2,2,1]=>3 [4,3,3,3]=>1 [4,3,3,2,1]=>5 [3,3,3,3,1]=>2 [3,3,3,2,2]=>1 [3,3,2,2,2,1]=>4 [9,5]=>1 [8,5,1]=>2 [7,5,2]=>1 [7,4,3]=>1 [6,4,4]=>1 [6,2,2,2,2]=>1 [5,5,4]=>1 [5,5,1,1,1,1]=>2 [5,4,3,2]=>1 [5,4,3,1,1]=>2 [5,4,2,2,1]=>3 [5,4,2,1,1,1]=>3 [5,3,3,2,1]=>4 [5,3,2,2,2]=>1 [5,2,2,2,2,1]=>3 [4,4,4,2]=>1 [4,4,3,3]=>1 [4,4,3,2,1]=>5 [4,3,2,2,2,1]=>5 [3,3,3,3,2]=>1 [3,3,3,3,1,1]=>2 [9,5,1]=>2 [8,5,2]=>1 [7,5,3]=>1 [6,5,4]=>1 [6,5,1,1,1,1]=>2 [6,3,3,3]=>1 [6,2,2,2,2,1]=>3 [5,5,5]=>1 [5,4,3,2,1]=>6 [5,4,3,1,1,1]=>2 [5,3,2,2,2,1]=>4 [4,4,4,3]=>1 [4,4,4,1,1,1]=>2 [3,3,3,3,3]=>1 [3,3,3,3,2,1]=>4 [8,5,3]=>1 [7,5,3,1]=>2 [5,5,3,3]=>1 [5,5,2,2,2]=>1 [5,4,3,2,1,1]=>6 [5,4,2,2,2,1]=>3 [4,4,4,4]=>1 [4,4,4,2,2]=>1 [4,3,3,3,2,1]=>5 [8,6,3]=>1 [6,5,3,3]=>1 [6,5,2,2,2]=>1 [6,4,4,3]=>1 [6,4,4,1,1,1]=>2 [6,3,3,3,2]=>1 [6,3,3,3,1,1]=>2 [5,5,4,3]=>1 [5,5,4,1,1,1]=>2 [5,5,2,2,2,1]=>3 [5,4,3,2,2,1]=>6 [5,3,3,3,2,1]=>4 [4,4,4,3,2]=>1 [4,4,4,3,1,1]=>2 [4,4,4,2,2,1]=>3 [4,4,4,3,2,1]=>5 [5,4,3,3,2,1]=>6 [6,3,3,3,2,1]=>4 [6,5,2,2,2,1]=>3 [5,5,3,3,1,1]=>2 [6,5,4,1,1,1]=>2 [5,5,3,3,2]=>1 [5,5,4,2,2]=>1 [6,4,4,2,2]=>1 [6,5,4,3]=>1 [9,6,3]=>1 [8,6,4]=>1 [5,4,4,3,2,1]=>6 [5,5,3,3,2,1]=>4 [5,5,4,2,2,1]=>3 [6,4,4,2,2,1]=>3 [5,5,4,3,1,1]=>2 [6,4,4,3,1,1]=>2 [6,5,3,3,1,1]=>2 [5,5,4,3,2]=>1 [6,4,4,3,2]=>1 [6,5,3,3,2]=>1 [6,5,4,2,2]=>1 [6,5,4,3,1]=>2 [6,5,4,1,1,1,1]=>2 [9,6,4]=>1 [8,5,4,2]=>1 [8,5,5,1]=>2 [5,5,4,3,2,1]=>6 [6,4,4,3,2,1]=>5 [6,5,3,3,2,1]=>4 [6,5,4,2,2,1]=>3 [6,5,4,3,1,1]=>2 [6,5,4,3,2]=>1 [6,5,2,2,2,2,1]=>3 [6,5,4,2,1,1,1]=>3 [7,5,4,3,1]=>2 [8,6,4,2]=>1 [10,6,4]=>1 [10,7,3]=>1 [9,7,4]=>1 [9,5,5,1]=>2 [6,5,4,3,2,1]=>7 [6,3,3,3,3,2,1]=>4 [6,5,3,2,2,2,1]=>4 [6,5,4,3,1,1,1]=>2 [11,7,3]=>1 [4,4,4,4,3,2,1]=>5 [6,4,3,3,3,2,1]=>5 [6,5,4,2,2,2,1]=>3 [6,5,4,3,2,1,1]=>7 [9,6,4,3]=>1 [5,4,4,4,3,2,1]=>6 [6,5,3,3,3,2,1]=>4 [6,5,4,3,2,2,1]=>7 [9,6,5,3]=>1 [8,6,5,3,1]=>2 [6,4,4,4,3,2,1]=>5 [6,5,4,3,3,2,1]=>7 [11,7,5,1]=>2 [9,7,5,3]=>1 [5,5,5,4,3,2,1]=>6 [6,5,4,4,3,2,1]=>7 [9,7,5,3,1]=>2 [10,7,5,3]=>1 [6,5,5,4,3,2,1]=>7 [9,7,5,4,1]=>2 [6,6,5,4,3,2,1]=>7 [7,6,5,4,3,2]=>1 [7,6,5,4,3,2,1]=>8 [7,6,5,4,3,1,1,1]=>2 [10,7,6,4,1]=>2 [9,7,6,4,2]=>1 [10,8,5,4,1]=>2 [7,6,5,4,3,2,1,1]=>8 [7,6,5,4,2,2,2,1]=>3 [10,8,6,4,1]=>2 [9,7,5,5,3,1]=>2 [7,6,5,4,3,2,2,1]=>8 [7,6,5,3,3,3,2,1]=>4 [11,8,6,4,1]=>2 [10,8,6,4,2]=>1 [7,6,5,4,3,3,2,1]=>8 [7,6,4,4,4,3,2,1]=>5 [11,8,6,5,1]=>2 [7,6,5,4,4,3,2,1]=>8 [7,5,5,5,4,3,2,1]=>6 [7,6,5,5,4,3,2,1]=>8 [6,6,6,5,4,3,2,1]=>7 [7,6,6,5,4,3,2,1]=>8 [12,9,7,5,1]=>2 [7,7,6,5,4,3,2,1]=>8 [13,9,7,5,1]=>2 [11,9,7,5,3,1]=>2 [11,8,7,5,4,1]=>2 [8,7,6,5,4,3,2,1]=>9 [8,7,6,5,4,3,2,1,1]=>9 [8,7,6,5,4,3,2,2,1]=>9 [8,7,6,5,4,3,3,2,1]=>9 [8,7,6,5,4,4,3,2,1]=>9 [11,9,7,5,5,3]=>1 [8,7,6,5,5,4,3,2,1]=>9 [8,7,6,6,5,4,3,2,1]=>9 [8,7,7,6,5,4,3,2,1]=>9 [8,8,7,6,5,4,3,2,1]=>9 [9,8,7,6,5,4,3,2,1]=>10 [11,9,7,7,5,3,3]=>1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].
References
[1] Hitczenko, Paweł, Louchard, G. Distinctness of compositions of an integer: a probabilistic analysis MathSciNet:1871561
[2] Triangle read by rows: T(n,k) is the number of partitions of n having least gap k. OEIS:A264401
[3] Hopkins, B., Sellers, J. A., Yee, A. J. Combinatorial Perspectives on the Crank and Mex Partition Statistics arXiv:2108.09414
Code
def statistic(pi):
    return min(set(range(1,len(pi)+2)).difference(set(pi)))
Created
Apr 08, 2017 at 22:40 by Martin Rubey
Updated
Aug 24, 2021 at 12:59 by Martin Rubey