- FindStat

Identifier

St000319: Integer partitions ⟶ 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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + q + q^{2}$

$F_{4} = 1 + 2\ q + q^{2} + q^{3}$

$F_{5} = 1 + 2\ q + 2\ q^{2} + q^{3} + q^{4}$

$F_{6} = 1 + 3\ q + 2\ q^{2} + 3\ q^{3} + q^{4} + q^{5}$

$F_{7} = 1 + 3\ q + 3\ q^{2} + 3\ q^{3} + 3\ q^{4} + q^{5} + q^{6}$

$F_{8} = 1 + 4\ q + 3\ q^{2} + 5\ q^{3} + 3\ q^{4} + 4\ q^{5} + q^{6} + q^{7}$

$F_{9} = 1 + 4\ q + 4\ q^{2} + 6\ q^{3} + 5\ q^{4} + 4\ q^{5} + 4\ q^{6} + q^{7} + q^{8}$

$F_{10} = 1 + 5\ q + 4\ q^{2} + 8\ q^{3} + 6\ q^{4} + 7\ q^{5} + 4\ q^{6} + 5\ q^{7} + q^{8} + q^{9}$

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