- FindStat

edit this statistic or download as text // json

Identifier

St000695: Set partitions ⟶ ℤ

Values

{{1,2}} => 1

{{1},{2}} => 1

{{1,2,3}} => 1

{{1,2},{3}} => 1

{{1,3},{2}} => 2

{{1},{2,3}} => 1

{{1},{2},{3}} => 1

{{1,2,3,4}} => 1

{{1,2,3},{4}} => 1

{{1,2,4},{3}} => 2

{{1,2},{3,4}} => 1

{{1,2},{3},{4}} => 1

{{1,3,4},{2}} => 2

{{1,3},{2,4}} => 2

{{1,3},{2},{4}} => 2

{{1,4},{2,3}} => 2

{{1},{2,3,4}} => 1

{{1},{2,3},{4}} => 1

{{1,4},{2},{3}} => 3

{{1},{2,4},{3}} => 1

{{1},{2},{3,4}} => 1

{{1},{2},{3},{4}} => 1

{{1,2,3,4,5}} => 1

{{1,2,3,4},{5}} => 1

{{1,2,3,5},{4}} => 2

{{1,2,3},{4,5}} => 1

{{1,2,3},{4},{5}} => 1

{{1,2,4,5},{3}} => 2

{{1,2,4},{3,5}} => 2

{{1,2,4},{3},{5}} => 2

{{1,2,5},{3,4}} => 2

{{1,2},{3,4,5}} => 1

{{1,2},{3,4},{5}} => 1

{{1,2,5},{3},{4}} => 3

{{1,2},{3,5},{4}} => 1

{{1,2},{3},{4,5}} => 1

{{1,2},{3},{4},{5}} => 1

{{1,3,4,5},{2}} => 2

{{1,3,4},{2,5}} => 2

{{1,3,4},{2},{5}} => 2

{{1,3,5},{2,4}} => 2

{{1,3},{2,4,5}} => 2

{{1,3},{2,4},{5}} => 2

{{1,3,5},{2},{4}} => 3

{{1,3},{2,5},{4}} => 3

{{1,3},{2},{4,5}} => 2

{{1,3},{2},{4},{5}} => 2

{{1,4,5},{2,3}} => 2

{{1,4},{2,3,5}} => 2

{{1,4},{2,3},{5}} => 2

{{1,5},{2,3,4}} => 2

{{1},{2,3,4,5}} => 1

{{1},{2,3,4},{5}} => 1

{{1,5},{2,3},{4}} => 3

{{1},{2,3,5},{4}} => 1

{{1},{2,3},{4,5}} => 1

{{1},{2,3},{4},{5}} => 1

{{1,4,5},{2},{3}} => 3

{{1,4},{2,5},{3}} => 3

{{1,4},{2},{3,5}} => 3

{{1,4},{2},{3},{5}} => 3

{{1,5},{2,4},{3}} => 3

{{1},{2,4,5},{3}} => 1

{{1},{2,4},{3,5}} => 1

{{1},{2,4},{3},{5}} => 1

{{1,5},{2},{3,4}} => 3

{{1},{2,5},{3,4}} => 1

{{1},{2},{3,4,5}} => 1

{{1},{2},{3,4},{5}} => 1

{{1,5},{2},{3},{4}} => 4

{{1},{2,5},{3},{4}} => 1

{{1},{2},{3,5},{4}} => 1

{{1},{2},{3},{4,5}} => 1

{{1},{2},{3},{4},{5}} => 1

{{1,2,3,4,5,6}} => 1

{{1,2,3,4,5},{6}} => 1

{{1,2,3,4,6},{5}} => 2

{{1,2,3,4},{5,6}} => 1

{{1,2,3,4},{5},{6}} => 1

{{1,2,3,5,6},{4}} => 2

{{1,2,3,5},{4,6}} => 2

{{1,2,3,5},{4},{6}} => 2

{{1,2,3,6},{4,5}} => 2

{{1,2,3},{4,5,6}} => 1

{{1,2,3},{4,5},{6}} => 1

{{1,2,3,6},{4},{5}} => 3

{{1,2,3},{4,6},{5}} => 1

{{1,2,3},{4},{5,6}} => 1

{{1,2,3},{4},{5},{6}} => 1

{{1,2,4,5,6},{3}} => 2

{{1,2,4,5},{3,6}} => 2

{{1,2,4,5},{3},{6}} => 2

{{1,2,4,6},{3,5}} => 2

{{1,2,4},{3,5,6}} => 2

{{1,2,4},{3,5},{6}} => 2

{{1,2,4,6},{3},{5}} => 3

{{1,2,4},{3,6},{5}} => 3

{{1,2,4},{3},{5,6}} => 2

{{1,2,4},{3},{5},{6}} => 2

{{1,2,5,6},{3,4}} => 2

{{1,2,5},{3,4,6}} => 2

>>> Load all 1200 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 number of blocks in the first part of the atomic decomposition of a set partition.
Let $\pi=(b_1,\dots,b_k)$ be a set partition with $k$ blocks, such that $\min b_i < \min b_{i+1}$. Then this statistic is the smallest number $\ell$ such that the union of the first $\ell$ blocks $b_1\cup\dots\cup b_\ell$ is an interval $\{1,\dots,m\}$.
The analogue for the decomposition of permutations is St000501The size of the first part in the decomposition of a permutation..

References

[1] Bergeron, N., Zabrocki, M. The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree arXiv:math/0509265

Code

def statistic(pi):
    """
    sage: pi = SetPartition([[1,7],[2,3,5],[4],[6,8]])
    sage: statistic(pi)
    4
    sage: pi = SetPartition([[1,2],[3,4,6],[5,7],[8]])
    sage: statistic(pi)
    1
    """
    blocks = sorted(sorted(b) for b in pi)
    u = []
    for i, b in enumerate(blocks, 1):
        u.extend(b)
        if max(u) == len(u):
            return i

Created

Feb 09, 2017 at 22:24 by Martin Rubey

Updated

Feb 09, 2017 at 22:24 by Martin Rubey