Identifier
Values
[1] => 1
[1,1] => 1
[2] => 1
[1,1,1] => 1
[1,2] => 2
[2,1] => 2
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 2
[1,2,1] => 3
[1,3] => 2
[2,1,1] => 2
[2,2] => 1
[3,1] => 2
[4] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 2
[1,1,2,1] => 3
[1,1,3] => 2
[1,2,1,1] => 3
[1,2,2] => 2
[1,3,1] => 3
[1,4] => 2
[2,1,1,1] => 2
[2,1,2] => 3
[2,2,1] => 2
[2,3] => 2
[3,1,1] => 2
[3,2] => 2
[4,1] => 2
[5] => 1
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 2
[1,1,1,2,1] => 3
[1,1,1,3] => 2
[1,1,2,1,1] => 3
[1,1,2,2] => 2
[1,1,3,1] => 3
[1,1,4] => 2
[1,2,1,1,1] => 3
[1,2,1,2] => 4
[1,2,2,1] => 3
[1,2,3] => 3
[1,3,1,1] => 3
[1,3,2] => 3
[1,4,1] => 3
[1,5] => 2
[2,1,1,1,1] => 2
[2,1,1,2] => 3
[2,1,2,1] => 4
[2,1,3] => 3
[2,2,1,1] => 2
[2,2,2] => 1
[2,3,1] => 3
[2,4] => 2
[3,1,1,1] => 2
[3,1,2] => 3
[3,2,1] => 3
[3,3] => 1
[4,1,1] => 2
[4,2] => 2
[5,1] => 2
[6] => 1
[1,1,1,1,1,1,1] => 1
[1,1,1,1,1,2] => 2
[1,1,1,1,2,1] => 3
[1,1,1,1,3] => 2
[1,1,1,2,1,1] => 3
[1,1,1,2,2] => 2
[1,1,1,3,1] => 3
[1,1,1,4] => 2
[1,1,2,1,1,1] => 3
[1,1,2,1,2] => 4
[1,1,2,2,1] => 3
[1,1,2,3] => 3
[1,1,3,1,1] => 3
[1,1,3,2] => 3
[1,1,4,1] => 3
[1,1,5] => 2
[1,2,1,1,1,1] => 3
[1,2,1,1,2] => 4
[1,2,1,2,1] => 5
[1,2,1,3] => 4
[1,2,2,1,1] => 3
[1,2,2,2] => 2
[1,2,3,1] => 4
[1,2,4] => 3
[1,3,1,1,1] => 3
[1,3,1,2] => 4
[1,3,2,1] => 4
[1,3,3] => 2
[1,4,1,1] => 3
[1,4,2] => 3
[1,5,1] => 3
[1,6] => 2
[2,1,1,1,1,1] => 2
[2,1,1,1,2] => 3
[2,1,1,2,1] => 4
[2,1,1,3] => 3
[2,1,2,1,1] => 4
[2,1,2,2] => 3
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Description
The number of runs in an integer composition.
Writing the composition as $c_1^{e_1} \dots c_\ell^{e_\ell}$, where $c_i \neq c_{i+1}$ for all $i$, the number of runs is $\ell$, see [def.2.8, 1].
It turns out that the total number of runs in all compositions of $n$ equals the total number of odd parts in all these compositions, see [1].
Writing the composition as $c_1^{e_1} \dots c_\ell^{e_\ell}$, where $c_i \neq c_{i+1}$ for all $i$, the number of runs is $\ell$, see [def.2.8, 1].
It turns out that the total number of runs in all compositions of $n$ equals the total number of odd parts in all these compositions, see [1].
References
[1] Hopkins, Brian, Mark Shattuck, and Andrew V. Sills. "Parts and subword patterns in compositions." http://home.dimacs.rutgers.edu/~asills/Comps/HopkinsSillsThanatipanondaWang0920AAM.pdf
Code
def runs(w):
i = 1
e = w[0]
for f in w[1:]:
if f != e:
i += 1
e = f
return i
Created
Apr 09, 2017 at 15:51 by Martin Rubey
Updated
Nov 19, 2017 at 22:05 by Christian Stump
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