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*       Copyright (C) 2019 The FindStatCrew <info@findstat.org>             *
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-----------------------------------------------------------------------------
Statistic identifier: St000160

-----------------------------------------------------------------------------
Collection: Integer partitions

-----------------------------------------------------------------------------
Description: The multiplicity of the smallest part of a partition.

This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.

The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].

-----------------------------------------------------------------------------
References: [1]   Andrews, G. E. The number of smallest parts in the partitions of $n$ [[MathSciNet:2456627]]
[2]   Chen, W. Y. C., Ji, K. Q., Zang, W. J. T. The spt-Crank for Ordinary Partitions [[arXiv:1308.3012]]

-----------------------------------------------------------------------------
Code:
def statistic(L):
    if not L:
        return 0
    return list(L).count(L[-1])

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

[]                        => 0
[1]                       => 1
[2]                       => 1
[1,1]                     => 2
[3]                       => 1
[2,1]                     => 1
[1,1,1]                   => 3
[4]                       => 1
[3,1]                     => 1
[2,2]                     => 2
[2,1,1]                   => 2
[1,1,1,1]                 => 4
[5]                       => 1
[4,1]                     => 1
[3,2]                     => 1
[3,1,1]                   => 2
[2,2,1]                   => 1
[2,1,1,1]                 => 3
[1,1,1,1,1]               => 5
[6]                       => 1
[5,1]                     => 1
[4,2]                     => 1
[4,1,1]                   => 2
[3,3]                     => 2
[3,2,1]                   => 1
[3,1,1,1]                 => 3
[2,2,2]                   => 3
[2,2,1,1]                 => 2
[2,1,1,1,1]               => 4
[1,1,1,1,1,1]             => 6
[7]                       => 1
[6,1]                     => 1
[5,2]                     => 1
[5,1,1]                   => 2
[4,3]                     => 1
[4,2,1]                   => 1
[4,1,1,1]                 => 3
[3,3,1]                   => 1
[3,2,2]                   => 2
[3,2,1,1]                 => 2
[3,1,1,1,1]               => 4
[2,2,2,1]                 => 1
[2,2,1,1,1]               => 3
[2,1,1,1,1,1]             => 5
[1,1,1,1,1,1,1]           => 7
[8]                       => 1
[7,1]                     => 1
[6,2]                     => 1
[6,1,1]                   => 2
[5,3]                     => 1
[5,2,1]                   => 1
[5,1,1,1]                 => 3
[4,4]                     => 2
[4,3,1]                   => 1
[4,2,2]                   => 2
[4,2,1,1]                 => 2
[4,1,1,1,1]               => 4
[3,3,2]                   => 1
[3,3,1,1]                 => 2
[3,2,2,1]                 => 1
[3,2,1,1,1]               => 3
[3,1,1,1,1,1]             => 5
[2,2,2,2]                 => 4
[2,2,2,1,1]               => 2
[2,2,1,1,1,1]             => 4
[2,1,1,1,1,1,1]           => 6
[1,1,1,1,1,1,1,1]         => 8
[9]                       => 1
[8,1]                     => 1
[7,2]                     => 1
[7,1,1]                   => 2
[6,3]                     => 1
[6,2,1]                   => 1
[6,1,1,1]                 => 3
[5,4]                     => 1
[5,3,1]                   => 1
[5,2,2]                   => 2
[5,2,1,1]                 => 2
[5,1,1,1,1]               => 4
[4,4,1]                   => 1
[4,3,2]                   => 1
[4,3,1,1]                 => 2
[4,2,2,1]                 => 1
[4,2,1,1,1]               => 3
[4,1,1,1,1,1]             => 5
[3,3,3]                   => 3
[3,3,2,1]                 => 1
[3,3,1,1,1]               => 3
[3,2,2,2]                 => 3
[3,2,2,1,1]               => 2
[3,2,1,1,1,1]             => 4
[3,1,1,1,1,1,1]           => 6
[2,2,2,2,1]               => 1
[2,2,2,1,1,1]             => 3
[2,2,1,1,1,1,1]           => 5
[2,1,1,1,1,1,1,1]         => 7
[1,1,1,1,1,1,1,1,1]       => 9
[10]                      => 1
[9,1]                     => 1
[8,2]                     => 1
[8,1,1]                   => 2
[7,3]                     => 1
[7,2,1]                   => 1
[7,1,1,1]                 => 3
[6,4]                     => 1
[6,3,1]                   => 1
[6,2,2]                   => 2
[6,2,1,1]                 => 2
[6,1,1,1,1]               => 4
[5,5]                     => 2
[5,4,1]                   => 1
[5,3,2]                   => 1
[5,3,1,1]                 => 2
[5,2,2,1]                 => 1
[5,2,1,1,1]               => 3
[5,1,1,1,1,1]             => 5
[4,4,2]                   => 1
[4,4,1,1]                 => 2
[4,3,3]                   => 2
[4,3,2,1]                 => 1
[4,3,1,1,1]               => 3
[4,2,2,2]                 => 3
[4,2,2,1,1]               => 2
[4,2,1,1,1,1]             => 4
[4,1,1,1,1,1,1]           => 6
[3,3,3,1]                 => 1
[3,3,2,2]                 => 2
[3,3,2,1,1]               => 2
[3,3,1,1,1,1]             => 4
[3,2,2,2,1]               => 1
[3,2,2,1,1,1]             => 3
[3,2,1,1,1,1,1]           => 5
[3,1,1,1,1,1,1,1]         => 7
[2,2,2,2,2]               => 5
[2,2,2,2,1,1]             => 2
[2,2,2,1,1,1,1]           => 4
[2,2,1,1,1,1,1,1]         => 6
[2,1,1,1,1,1,1,1,1]       => 8
[1,1,1,1,1,1,1,1,1,1]     => 10
[11]                      => 1
[10,1]                    => 1
[9,2]                     => 1
[9,1,1]                   => 2
[8,3]                     => 1
[8,2,1]                   => 1
[8,1,1,1]                 => 3
[7,4]                     => 1
[7,3,1]                   => 1
[7,2,2]                   => 2
[7,2,1,1]                 => 2
[7,1,1,1,1]               => 4
[6,5]                     => 1
[6,4,1]                   => 1
[6,3,2]                   => 1
[6,3,1,1]                 => 2
[6,2,2,1]                 => 1
[6,2,1,1,1]               => 3
[6,1,1,1,1,1]             => 5
[5,5,1]                   => 1
[5,4,2]                   => 1
[5,4,1,1]                 => 2
[5,3,3]                   => 2
[5,3,2,1]                 => 1
[5,3,1,1,1]               => 3
[5,2,2,2]                 => 3
[5,2,2,1,1]               => 2
[5,2,1,1,1,1]             => 4
[5,1,1,1,1,1,1]           => 6
[4,4,3]                   => 1
[4,4,2,1]                 => 1
[4,4,1,1,1]               => 3
[4,3,3,1]                 => 1
[4,3,2,2]                 => 2
[4,3,2,1,1]               => 2
[4,3,1,1,1,1]             => 4
[4,2,2,2,1]               => 1
[4,2,2,1,1,1]             => 3
[4,2,1,1,1,1,1]           => 5
[4,1,1,1,1,1,1,1]         => 7
[3,3,3,2]                 => 1
[3,3,3,1,1]               => 2
[3,3,2,2,1]               => 1
[3,3,2,1,1,1]             => 3
[3,3,1,1,1,1,1]           => 5
[3,2,2,2,2]               => 4
[3,2,2,2,1,1]             => 2
[3,2,2,1,1,1,1]           => 4
[3,2,1,1,1,1,1,1]         => 6
[3,1,1,1,1,1,1,1,1]       => 8
[2,2,2,2,2,1]             => 1
[2,2,2,2,1,1,1]           => 3
[2,2,2,1,1,1,1,1]         => 5
[2,2,1,1,1,1,1,1,1]       => 7
[2,1,1,1,1,1,1,1,1,1]     => 9
[1,1,1,1,1,1,1,1,1,1,1]   => 11
[12]                      => 1
[11,1]                    => 1
[10,2]                    => 1
[10,1,1]                  => 2
[9,3]                     => 1
[9,2,1]                   => 1
[9,1,1,1]                 => 3
[8,4]                     => 1
[8,3,1]                   => 1
[8,2,2]                   => 2
[8,2,1,1]                 => 2
[8,1,1,1,1]               => 4
[7,5]                     => 1
[7,4,1]                   => 1
[7,3,2]                   => 1
[7,3,1,1]                 => 2
[7,2,2,1]                 => 1
[7,2,1,1,1]               => 3
[7,1,1,1,1,1]             => 5
[6,6]                     => 2
[6,5,1]                   => 1
[6,4,2]                   => 1
[6,4,1,1]                 => 2
[6,3,3]                   => 2
[6,3,2,1]                 => 1
[6,3,1,1,1]               => 3
[6,2,2,2]                 => 3
[6,2,2,1,1]               => 2
[6,2,1,1,1,1]             => 4
[6,1,1,1,1,1,1]           => 6
[5,5,2]                   => 1
[5,5,1,1]                 => 2
[5,4,3]                   => 1
[5,4,2,1]                 => 1
[5,4,1,1,1]               => 3
[5,3,3,1]                 => 1
[5,3,2,2]                 => 2
[5,3,2,1,1]               => 2
[5,3,1,1,1,1]             => 4
[5,2,2,2,1]               => 1
[5,2,2,1,1,1]             => 3
[5,2,1,1,1,1,1]           => 5
[5,1,1,1,1,1,1,1]         => 7
[4,4,4]                   => 3
[4,4,3,1]                 => 1
[4,4,2,2]                 => 2
[4,4,2,1,1]               => 2
[4,4,1,1,1,1]             => 4
[4,3,3,2]                 => 1
[4,3,3,1,1]               => 2
[4,3,2,2,1]               => 1
[4,3,2,1,1,1]             => 3
[4,3,1,1,1,1,1]           => 5
[4,2,2,2,2]               => 4
[4,2,2,2,1,1]             => 2
[4,2,2,1,1,1,1]           => 4
[4,2,1,1,1,1,1,1]         => 6
[4,1,1,1,1,1,1,1,1]       => 8
[3,3,3,3]                 => 4
[3,3,3,2,1]               => 1
[3,3,3,1,1,1]             => 3
[3,3,2,2,2]               => 3
[3,3,2,2,1,1]             => 2
[3,3,2,1,1,1,1]           => 4
[3,3,1,1,1,1,1,1]         => 6
[3,2,2,2,2,1]             => 1
[3,2,2,2,1,1,1]           => 3
[3,2,2,1,1,1,1,1]         => 5
[3,2,1,1,1,1,1,1,1]       => 7
[3,1,1,1,1,1,1,1,1,1]     => 9
[2,2,2,2,2,2]             => 6
[2,2,2,2,2,1,1]           => 2
[2,2,2,2,1,1,1,1]         => 4
[2,2,2,1,1,1,1,1,1]       => 6
[2,2,1,1,1,1,1,1,1,1]     => 8
[2,1,1,1,1,1,1,1,1,1,1]   => 10
[1,1,1,1,1,1,1,1,1,1,1,1] => 12
[5,4,3,1]                 => 1
[5,4,2,2]                 => 2
[5,4,2,1,1]               => 2
[5,3,3,2]                 => 1
[5,3,3,1,1]               => 2
[5,3,2,2,1]               => 1
[4,4,3,2]                 => 1
[4,4,3,1,1]               => 2
[4,4,2,2,1]               => 1
[4,3,3,2,1]               => 1
[5,4,3,2]                 => 1
[5,4,3,1,1]               => 2
[5,4,2,2,1]               => 1
[5,3,3,2,1]               => 1
[4,4,3,2,1]               => 1
[5,4,3,2,1]               => 1

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Created: Sep 04, 2013 at 14:19 by Christian Stump

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Last Updated: Mar 31, 2019 at 21:51 by Martin Rubey