Identifier
- St001928: Permutations ⟶ ℤ
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 1
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 1
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
[1,5,2,3,4] => 1
[1,5,2,4,3] => 2
[1,5,3,2,4] => 1
[1,5,3,4,2] => 2
[1,5,4,2,3] => 1
[1,5,4,3,2] => 2
[2,1,3,4,5] => 1
[2,1,3,5,4] => 2
[2,1,4,3,5] => 2
[2,1,4,5,3] => 2
[2,1,5,3,4] => 2
[2,1,5,4,3] => 2
[2,3,1,4,5] => 1
[2,3,1,5,4] => 2
[2,3,4,1,5] => 1
[2,3,4,5,1] => 1
[2,3,5,1,4] => 1
[2,3,5,4,1] => 1
[2,4,1,3,5] => 1
[2,4,1,5,3] => 2
[2,4,3,1,5] => 1
[2,4,3,5,1] => 2
[2,4,5,1,3] => 1
[2,4,5,3,1] => 1
[2,5,1,3,4] => 1
[2,5,1,4,3] => 2
[2,5,3,1,4] => 1
[2,5,3,4,1] => 2
[2,5,4,1,3] => 1
[2,5,4,3,1] => 2
[3,1,2,4,5] => 1
[3,1,2,5,4] => 2
[3,1,4,2,5] => 2
[3,1,4,5,2] => 2
[3,1,5,2,4] => 2
[3,1,5,4,2] => 2
[3,2,1,4,5] => 1
[3,2,1,5,4] => 2
[3,2,4,1,5] => 2
[3,2,4,5,1] => 2
[3,2,5,1,4] => 2
[3,2,5,4,1] => 2
[3,4,1,2,5] => 1
[3,4,1,5,2] => 2
[3,4,2,1,5] => 1
[3,4,2,5,1] => 2
[3,4,5,1,2] => 1
[3,4,5,2,1] => 1
[3,5,1,2,4] => 1
[3,5,1,4,2] => 2
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Description
The number of non-overlapping descents in a permutation.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
References
[1] Kitaev, S., Zhang, P. B. Non-overlapping descents and ascents in stack-sortable permutations arXiv:2310.17236
Code
def statistic(pi):
p = Permutation([2,1])
from sage.combinat.permutation import to_standard
k = len(p)
i = 0
c = 0
while i <= len(pi) - k:
if to_standard(pi[i:i+k]) == p:
c += 1
i += k
else:
i += 1
return c
def generating_function(n):
P. = QQ[]
L. = LazyPowerSeriesRing(P)
gf = exp(t)/(1-x*(1+(t-1)*exp(t)))
return factorial(n)*gf[n]
Created
Oct 28, 2023 at 19:47 by Martin Rubey
Updated
Oct 28, 2023 at 19:47 by Martin Rubey
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