- FindStat

Identifier

St000160: Integer partitions ⟶ ℤ

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

>>> Load all 288 entries. <<<

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 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])

Created

Sep 04, 2013 at 14:19 by Christian Stump

Updated

Mar 31, 2019 at 21:51 by Martin Rubey