- FindStat

Identifier

St000319: Integer partitions ⟶ ℤ (values match St000320The dinv adjustment of an integer partition.)

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [1]=>0
[2]=>1
[1,1]=>0
[3]=>2
[2,1]=>1
[1,1,1]=>0
[4]=>3
[3,1]=>2
[2,2]=>1
[2,1,1]=>1
[1,1,1,1]=>0
[5]=>4
[4,1]=>3
[3,2]=>2
[3,1,1]=>2
[2,2,1]=>1
[2,1,1,1]=>1
[1,1,1,1,1]=>0
[6]=>5
[5,1]=>4
[4,2]=>3
[4,1,1]=>3
[3,3]=>3
[3,2,1]=>2
[3,1,1,1]=>2
[2,2,2]=>1
[2,2,1,1]=>1
[2,1,1,1,1]=>1
[1,1,1,1,1,1]=>0
[7]=>6
[6,1]=>5
[5,2]=>4
[5,1,1]=>4
[4,3]=>4
[4,2,1]=>3
[4,1,1,1]=>3
[3,3,1]=>3
[3,2,2]=>2
[3,2,1,1]=>2
[3,1,1,1,1]=>2
[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]=>7
[7,1]=>6
[6,2]=>5
[6,1,1]=>5
[5,3]=>5
[5,2,1]=>4
[5,1,1,1]=>4
[4,4]=>5
[4,3,1]=>4
[4,2,2]=>3
[4,2,1,1]=>3
[4,1,1,1,1]=>3
[3,3,2]=>3
[3,3,1,1]=>3
[3,2,2,1]=>2
[3,2,1,1,1]=>2
[3,1,1,1,1,1]=>2
[2,2,2,2]=>1
[2,2,2,1,1]=>1
[2,2,1,1,1,1]=>1
[2,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1]=>0
[9]=>8
[8,1]=>7
[7,2]=>6
[7,1,1]=>6
[6,3]=>6
[6,2,1]=>5
[6,1,1,1]=>5
[5,4]=>6
[5,3,1]=>5
[5,2,2]=>4
[5,2,1,1]=>4
[5,1,1,1,1]=>4
[4,4,1]=>5
[4,3,2]=>4
[4,3,1,1]=>4
[4,2,2,1]=>3
[4,2,1,1,1]=>3
[4,1,1,1,1,1]=>3
[3,3,3]=>3
[3,3,2,1]=>3
[3,3,1,1,1]=>3
[3,2,2,2]=>2
[3,2,2,1,1]=>2
[3,2,1,1,1,1]=>2
[3,1,1,1,1,1,1]=>2
[2,2,2,2,1]=>1
[2,2,2,1,1,1]=>1
[2,2,1,1,1,1,1]=>1
[2,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1,1]=>0
[10]=>9
[9,1]=>8
[8,2]=>7
[8,1,1]=>7
[7,3]=>7
[7,2,1]=>6
[7,1,1,1]=>6
[6,4]=>7
[6,3,1]=>6
[6,2,2]=>5
[6,2,1,1]=>5
[6,1,1,1,1]=>5
[5,5]=>7
[5,4,1]=>6
[5,3,2]=>5
[5,3,1,1]=>5
[5,2,2,1]=>4
[5,2,1,1,1]=>4
[5,1,1,1,1,1]=>4
[4,4,2]=>5
[4,4,1,1]=>5
[4,3,3]=>4
[4,3,2,1]=>4
[4,3,1,1,1]=>4
[4,2,2,2]=>3
[4,2,2,1,1]=>3
[4,2,1,1,1,1]=>3
[4,1,1,1,1,1,1]=>3
[3,3,3,1]=>3
[3,3,2,2]=>3
[3,3,2,1,1]=>3
[3,3,1,1,1,1]=>3
[3,2,2,2,1]=>2
[3,2,2,1,1,1]=>2
[3,2,1,1,1,1,1]=>2
[3,1,1,1,1,1,1,1]=>2
[2,2,2,2,2]=>1
[2,2,2,2,1,1]=>1
[2,2,2,1,1,1,1]=>1
[2,2,1,1,1,1,1,1]=>1
[2,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1,1,1]=>0
                    

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Description

The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.

References

[1] , Loehr, N. A., Warrington, G. S. Nested quantum Dyck paths and $\nabla (s_\lambda )$ MathSciNet:2418288 arXiv:0705.4608
[2] Haglund, J. The $q$,$t$-Catalan numbers and the space of diagonal harmonics MathSciNet:2371044

Code

def remove_border_strip(L):
    return Partition( part-1 for part in L[1:] if part > 1 )

def border_strip_decomposition(L):
    decomp = []
    while len(L) > 0:
        decomp.append(L)
        L = remove_border_strip(L)
    return decomp

def border_strip_crossing_number(L):
    L = list(L)+[1]
    return sum( L[i-1]-L[i] for i in range(1,len(L)) )

def statistic(L):
    return sum( border_strip_crossing_number(X) for X in border_strip_decomposition(L) )

Created

Dec 08, 2015 at 17:15 by Christian Stump

Updated

Dec 17, 2015 at 11:19 by Christian Stump