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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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* This information is distributed in the hope that it will be useful, *
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-----------------------------------------------------------------------------
Statistic identifier: St000759
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
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)))
-----------------------------------------------------------------------------
Statistic values:
[] => 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
-----------------------------------------------------------------------------
Created: Apr 08, 2017 at 22:40 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Aug 24, 2021 at 12:59 by Martin Rubey