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Identifier

St000056: Permutations ⟶ ℤ (values match St000234The number of global ascents of a permutation.)

Values

[1] => 1

[1,2] => 2

[2,1] => 1

[1,2,3] => 3

[1,3,2] => 2

[2,1,3] => 2

[2,3,1] => 1

[3,1,2] => 1

[3,2,1] => 1

[1,2,3,4] => 4

[1,2,4,3] => 3

[1,3,2,4] => 3

[1,3,4,2] => 2

[1,4,2,3] => 2

[1,4,3,2] => 2

[2,1,3,4] => 3

[2,1,4,3] => 2

[2,3,1,4] => 2

[2,3,4,1] => 1

[2,4,1,3] => 1

[2,4,3,1] => 1

[3,1,2,4] => 2

[3,1,4,2] => 1

[3,2,1,4] => 2

[3,2,4,1] => 1

[3,4,1,2] => 1

[3,4,2,1] => 1

[4,1,2,3] => 1

[4,1,3,2] => 1

[4,2,1,3] => 1

[4,2,3,1] => 1

[4,3,1,2] => 1

[4,3,2,1] => 1

[1,2,3,4,5] => 5

[1,2,3,5,4] => 4

[1,2,4,3,5] => 4

[1,2,4,5,3] => 3

[1,2,5,3,4] => 3

[1,2,5,4,3] => 3

[1,3,2,4,5] => 4

[1,3,2,5,4] => 3

[1,3,4,2,5] => 3

[1,3,4,5,2] => 2

[1,3,5,2,4] => 2

[1,3,5,4,2] => 2

[1,4,2,3,5] => 3

[1,4,2,5,3] => 2

[1,4,3,2,5] => 3

[1,4,3,5,2] => 2

[1,4,5,2,3] => 2

[1,4,5,3,2] => 2

[1,5,2,3,4] => 2

[1,5,2,4,3] => 2

[1,5,3,2,4] => 2

[1,5,3,4,2] => 2

[1,5,4,2,3] => 2

[1,5,4,3,2] => 2

[2,1,3,4,5] => 4

[2,1,3,5,4] => 3

[2,1,4,3,5] => 3

[2,1,4,5,3] => 2

[2,1,5,3,4] => 2

[2,1,5,4,3] => 2

[2,3,1,4,5] => 3

[2,3,1,5,4] => 2

[2,3,4,1,5] => 2

[2,3,4,5,1] => 1

[2,3,5,1,4] => 1

[2,3,5,4,1] => 1

[2,4,1,3,5] => 2

[2,4,1,5,3] => 1

[2,4,3,1,5] => 2

[2,4,3,5,1] => 1

[2,4,5,1,3] => 1

[2,4,5,3,1] => 1

[2,5,1,3,4] => 1

[2,5,1,4,3] => 1

[2,5,3,1,4] => 1

[2,5,3,4,1] => 1

[2,5,4,1,3] => 1

[2,5,4,3,1] => 1

[3,1,2,4,5] => 3

[3,1,2,5,4] => 2

[3,1,4,2,5] => 2

[3,1,4,5,2] => 1

[3,1,5,2,4] => 1

[3,1,5,4,2] => 1

[3,2,1,4,5] => 3

[3,2,1,5,4] => 2

[3,2,4,1,5] => 2

[3,2,4,5,1] => 1

[3,2,5,1,4] => 1

[3,2,5,4,1] => 1

[3,4,1,2,5] => 2

[3,4,1,5,2] => 1

[3,4,2,1,5] => 2

[3,4,2,5,1] => 1

[3,4,5,1,2] => 1

[3,4,5,2,1] => 1

[3,5,1,2,4] => 1

[3,5,1,4,2] => 1

>>> Load all 873 entries. <<<

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Description

The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus St000234The number of global ascents of a permutation..

References

[1] Callan, D. Counting stabilized-interval-free permutations MathSciNet:2049703 arXiv:math/0310157
[2] Adin, R. M., Bagno, E., Roichman, Y. Block decomposition of permutations and Schur-positivity arXiv:1611.06979
[3] Stanley, R. P. The Descent Set and Connectivity Set of a Permutation arXiv:math/0507224

Code

def statistic(pi):
    return len([i for i in [1..pi.size()] if sorted(pi[j] for j in range(i)) == [1..i]])

Created

Mar 26, 2013 at 06:07 by Christian Stump

Updated

Jul 12, 2017 at 16:47 by Christian Stump