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Statistic identifier: St000320
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Collection: Integer partitions
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Description: The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
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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]]
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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 length_seq(L):
strip_seq = border_strip_decomposition(L) + [[]]
len_seq = [0]*(L[0])
for j in range(len(strip_seq)-1):
len_seq[L[0]-strip_seq[j][0]] = sum(strip_seq[j])-sum(strip_seq[j+1])
return len_seq
def statistic(L):
len_seq = length_seq(L)
return sum( L[0]-1-j for j in range(len(len_seq)) if len_seq[j] > 0 )
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Statistic 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
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Created: Dec 08, 2015 at 17:40 by Christian Stump
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Last Updated: Dec 17, 2015 at 12:53 by Christian Stump