Identifier
Identifier
• St000319: ⟶ ℤ (values match St000320The dinv adjustment of an integer partition.)
Values
[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
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