Processing math: 100%

Your data matches 1 statistic following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000749
(load all 2 compositions to match this statistic)
Mapping
St000749: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[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
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 S8 yields S(5,3)⊕S(6,2) of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to S7 yields S(4,3)⊕2S(5,2)⊕S(6,1) of degrees 14, 14 and 6. However, restricting to S6 yields S(3,3)⊕3S(4,2)⊕3S(5,1)⊕S6 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].