Identifier
Values
=>
Cc0002;cc-rep
[1]=>0
[2]=>0
[1,1]=>0
[3]=>0
[2,1]=>1
[1,1,1]=>0
[4]=>0
[3,1]=>0
[2,2]=>2
[2,1,1]=>0
[1,1,1,1]=>0
[5]=>0
[4,1]=>1
[3,2]=>0
[3,1,1]=>1
[2,2,1]=>0
[2,1,1,1]=>1
[1,1,1,1,1]=>0
[6]=>0
[5,1]=>0
[4,2]=>0
[4,1,1]=>2
[3,3]=>0
[3,2,1]=>1
[3,1,1,1]=>2
[2,2,2]=>0
[2,2,1,1]=>0
[2,1,1,1,1]=>0
[1,1,1,1,1,1]=>0
[7]=>0
[6,1]=>1
[5,2]=>1
[5,1,1]=>0
[4,3]=>1
[4,2,1]=>0
[4,1,1,1]=>3
[3,3,1]=>0
[3,2,2]=>0
[3,2,1,1]=>0
[3,1,1,1,1]=>0
[2,2,2,1]=>1
[2,2,1,1,1]=>1
[2,1,1,1,1,1]=>1
[1,1,1,1,1,1,1]=>0
[8]=>0
[7,1]=>0
[6,2]=>2
[6,1,1]=>0
[5,3]=>2
[5,2,1]=>1
[5,1,1,1]=>0
[4,4]=>2
[4,3,1]=>1
[4,2,2]=>1
[4,2,1,1]=>1
[4,1,1,1,1]=>0
[3,3,2]=>1
[3,3,1,1]=>1
[3,2,2,1]=>1
[3,2,1,1,1]=>1
[3,1,1,1,1,1]=>0
[2,2,2,2]=>2
[2,2,2,1,1]=>2
[2,2,1,1,1,1]=>2
[2,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,1,1]=>0
[9]=>0
[8,1]=>1
[7,2]=>0
[7,1,1]=>1
[6,3]=>3
[6,2,1]=>0
[6,1,1,1]=>1
[5,4]=>3
[5,3,1]=>2
[5,2,2]=>2
[5,2,1,1]=>0
[5,1,1,1,1]=>1
[4,4,1]=>2
[4,3,2]=>2
[4,3,1,1]=>2
[4,2,2,1]=>2
[4,2,1,1,1]=>0
[4,1,1,1,1,1]=>1
[3,3,3]=>2
[3,3,2,1]=>2
[3,3,1,1,1]=>2
[3,2,2,2]=>2
[3,2,2,1,1]=>2
[3,2,1,1,1,1]=>0
[3,1,1,1,1,1,1]=>1
[2,2,2,2,1]=>3
[2,2,2,1,1,1]=>3
[2,2,1,1,1,1,1]=>0
[2,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1,1]=>0
[10]=>0
[9,1]=>0
[8,2]=>0
[8,1,1]=>2
[7,3]=>0
[7,2,1]=>1
[7,1,1,1]=>2
[6,4]=>4
[6,3,1]=>0
[6,2,2]=>0
[6,2,1,1]=>1
[6,1,1,1,1]=>2
[5,5]=>4
[5,4,1]=>3
[5,3,2]=>3
[5,3,1,1]=>0
[5,2,2,1]=>0
[5,2,1,1,1]=>1
[5,1,1,1,1,1]=>2
[4,4,2]=>3
[4,4,1,1]=>3
[4,3,3]=>3
[4,3,2,1]=>3
[4,3,1,1,1]=>0
[4,2,2,2]=>3
[4,2,2,1,1]=>0
[4,2,1,1,1,1]=>1
[4,1,1,1,1,1,1]=>2
[3,3,3,1]=>3
[3,3,2,2]=>3
[3,3,2,1,1]=>3
[3,3,1,1,1,1]=>0
[3,2,2,2,1]=>3
[3,2,2,1,1,1]=>0
[3,2,1,1,1,1,1]=>1
[3,1,1,1,1,1,1,1]=>2
[2,2,2,2,2]=>4
[2,2,2,2,1,1]=>4
[2,2,2,1,1,1,1]=>0
[2,2,1,1,1,1,1,1]=>0
[2,1,1,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,1,1,1,1]=>0
[11]=>0
[10,1]=>1
[9,2]=>1
[9,1,1]=>0
[8,3]=>1
[8,2,1]=>0
[8,1,1,1]=>3
[7,4]=>0
[7,3,1]=>1
[7,2,2]=>0
[7,2,1,1]=>2
[7,1,1,1,1]=>3
[6,5]=>5
[6,4,1]=>0
[6,3,2]=>1
[6,3,1,1]=>1
[6,2,2,1]=>1
[6,2,1,1,1]=>2
[6,1,1,1,1,1]=>3
[5,5,1]=>4
[5,4,2]=>4
[5,4,1,1]=>0
[5,3,3]=>4
[5,3,2,1]=>1
[5,3,1,1,1]=>1
[5,2,2,2]=>0
[5,2,2,1,1]=>1
[5,2,1,1,1,1]=>2
[5,1,1,1,1,1,1]=>3
[4,4,3]=>4
[4,4,2,1]=>4
[4,4,1,1,1]=>0
[4,3,3,1]=>4
[4,3,2,2]=>4
[4,3,2,1,1]=>1
[4,3,1,1,1,1]=>1
[4,2,2,2,1]=>0
[4,2,2,1,1,1]=>1
[4,2,1,1,1,1,1]=>2
[4,1,1,1,1,1,1,1]=>3
[3,3,3,2]=>4
[3,3,3,1,1]=>4
[3,3,2,2,1]=>4
[3,3,2,1,1,1]=>1
[3,3,1,1,1,1,1]=>0
[3,2,2,2,2]=>4
[3,2,2,2,1,1]=>0
[3,2,2,1,1,1,1]=>1
[3,2,1,1,1,1,1,1]=>0
[3,1,1,1,1,1,1,1,1]=>0
[2,2,2,2,2,1]=>5
[2,2,2,2,1,1,1]=>0
[2,2,2,1,1,1,1,1]=>1
[2,2,1,1,1,1,1,1,1]=>1
[2,1,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1,1,1,1]=>0
[12]=>0
[11,1]=>0
[10,2]=>2
[10,1,1]=>0
[9,3]=>2
[9,2,1]=>1
[9,1,1,1]=>0
[8,4]=>0
[8,3,1]=>0
[8,2,2]=>1
[8,2,1,1]=>0
[8,1,1,1,1]=>4
[7,5]=>0
[7,4,1]=>1
[7,3,2]=>0
[7,3,1,1]=>2
[7,2,2,1]=>0
[7,2,1,1,1]=>3
[7,1,1,1,1,1]=>4
[6,6]=>6
[6,5,1]=>0
[6,4,2]=>0
[6,4,1,1]=>1
[6,3,3]=>2
[6,3,2,1]=>2
[6,3,1,1,1]=>2
[6,2,2,2]=>0
[6,2,2,1,1]=>2
[6,2,1,1,1,1]=>3
[6,1,1,1,1,1,1]=>4
[5,5,2]=>5
[5,5,1,1]=>0
[5,4,3]=>5
[5,4,2,1]=>0
[5,4,1,1,1]=>1
[5,3,3,1]=>2
[5,3,2,2]=>0
[5,3,2,1,1]=>2
[5,3,1,1,1,1]=>2
[5,2,2,2,1]=>1
[5,2,2,1,1,1]=>2
[5,2,1,1,1,1,1]=>3
[5,1,1,1,1,1,1,1]=>4
[4,4,4]=>5
[4,4,3,1]=>5
[4,4,2,2]=>5
[4,4,2,1,1]=>0
[4,4,1,1,1,1]=>0
[4,3,3,2]=>5
[4,3,3,1,1]=>2
[4,3,2,2,1]=>0
[4,3,2,1,1,1]=>2
[4,3,1,1,1,1,1]=>0
[4,2,2,2,2]=>0
[4,2,2,2,1,1]=>1
[4,2,2,1,1,1,1]=>2
[4,2,1,1,1,1,1,1]=>0
[4,1,1,1,1,1,1,1,1]=>0
[3,3,3,3]=>5
[3,3,3,2,1]=>5
[3,3,3,1,1,1]=>2
[3,3,2,2,2]=>5
[3,3,2,2,1,1]=>0
[3,3,2,1,1,1,1]=>0
[3,3,1,1,1,1,1,1]=>1
[3,2,2,2,2,1]=>0
[3,2,2,2,1,1,1]=>1
[3,2,2,1,1,1,1,1]=>0
[3,2,1,1,1,1,1,1,1]=>1
[3,1,1,1,1,1,1,1,1,1]=>0
[2,2,2,2,2,2]=>6
[2,2,2,2,2,1,1]=>0
[2,2,2,2,1,1,1,1]=>0
[2,2,2,1,1,1,1,1,1]=>2
[2,2,1,1,1,1,1,1,1,1]=>2
[2,1,1,1,1,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,1,1,1,1,1,1]=>0
[5,4,3,1]=>0
[5,4,2,2]=>1
[5,4,2,1,1]=>1
[5,3,3,2]=>0
[5,3,3,1,1]=>3
[5,3,2,2,1]=>1
[4,4,3,2]=>6
[4,4,3,1,1]=>0
[4,4,2,2,1]=>1
[4,3,3,2,1]=>0
[5,4,3,2]=>1
[5,4,3,1,1]=>1
[5,4,2,2,1]=>2
[5,3,3,2,1]=>1
[4,4,3,2,1]=>1
[5,4,3,2,1]=>2
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
Description
The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
References
[1] Irie, Y. p-Saturations of Welter's Game and the Irreducible Representations of Symmetric Groups arXiv:1604.07214
Code
def branching_symmetric_group(la, p):
"""
Return a dictionary from partitions to multiplicities.
"""
la = Partition(la)
l = {la: 1}
for i in range(la.size()-p):
l_new = dict()
for mu in l:
for r, _ in mu.removable_cells():
nu = mu.remove_cell(r)
l_new[nu] = l_new.get(nu, 0) + l[mu]
l = l_new
return l
def statistic(la):
"""Return the largest number such that the restriction of the
irreducible representation corresponding to la has a component
relative prime to 2.
"""
la = Partition(la)
for m in range(la.size(), 0, -1):
if any(gcd(StandardTableaux(mu).cardinality(), 2) == 1
for mu in branching_symmetric_group(la, m)):
return la.size()-m
Created
Apr 05, 2017 at 11:28 by Martin Rubey
Updated
Sep 14, 2018 at 18:56 by Martin Rubey
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!