edit this statistic or download as text // json
Identifier
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
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 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