Identifier
Values
=>
Cc0002;cc-rep
[]=>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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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].
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
[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])
Created
Sep 04, 2013 at 14:19 by Christian Stump
Updated
Mar 31, 2019 at 21:51 by Martin Rubey
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