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
[2,1,3,1]=>4
[2,1,4]=>3
[2,2,1,1,1]=>2
[2,2,1,2]=>3
[2,2,2,1]=>2
[2,2,3]=>2
[2,3,1,1]=>3
[2,3,2]=>3
[2,4,1]=>3
[2,5]=>2
[3,1,1,1,1]=>2
[3,1,1,2]=>3
[3,1,2,1]=>4
[3,1,3]=>3
[3,2,1,1]=>3
[3,2,2]=>2
[3,3,1]=>2
[3,4]=>2
[4,1,1,1]=>2
[4,1,2]=>3
[4,2,1]=>3
[4,3]=>2
[5,1,1]=>2
[5,2]=>2
[6,1]=>2
[7]=>1
[1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,2]=>2
[1,1,1,1,1,2,1]=>3
[1,1,1,1,1,3]=>2
[1,1,1,1,2,1,1]=>3
[1,1,1,1,2,2]=>2
[1,1,1,1,3,1]=>3
[1,1,1,1,4]=>2
[1,1,1,2,1,1,1]=>3
[1,1,1,2,1,2]=>4
[1,1,1,2,2,1]=>3
[1,1,1,2,3]=>3
[1,1,1,3,1,1]=>3
[1,1,1,3,2]=>3
[1,1,1,4,1]=>3
[1,1,1,5]=>2
[1,1,2,1,1,1,1]=>3
[1,1,2,1,1,2]=>4
[1,1,2,1,2,1]=>5
[1,1,2,1,3]=>4
[1,1,2,2,1,1]=>3
[1,1,2,2,2]=>2
[1,1,2,3,1]=>4
[1,1,2,4]=>3
[1,1,3,1,1,1]=>3
[1,1,3,1,2]=>4
[1,1,3,2,1]=>4
[1,1,3,3]=>2
[1,1,4,1,1]=>3
[1,1,4,2]=>3
[1,1,5,1]=>3
[1,1,6]=>2
[1,2,1,1,1,1,1]=>3
[1,2,1,1,1,2]=>4
[1,2,1,1,2,1]=>5
[1,2,1,1,3]=>4
[1,2,1,2,1,1]=>5
[1,2,1,2,2]=>4
[1,2,1,3,1]=>5
[1,2,1,4]=>4
[1,2,2,1,1,1]=>3
[1,2,2,1,2]=>4
[1,2,2,2,1]=>3
[1,2,2,3]=>3
[1,2,3,1,1]=>4
[1,2,3,2]=>4
[1,2,4,1]=>4
[1,2,5]=>3
[1,3,1,1,1,1]=>3
[1,3,1,1,2]=>4
[1,3,1,2,1]=>5
[1,3,1,3]=>4
[1,3,2,1,1]=>4
[1,3,2,2]=>3
[1,3,3,1]=>3
[1,3,4]=>3
[1,4,1,1,1]=>3
[1,4,1,2]=>4
[1,4,2,1]=>4
[1,4,3]=>3
[1,5,1,1]=>3
[1,5,2]=>3
[1,6,1]=>3
[1,7]=>2
[2,1,1,1,1,1,1]=>2
[2,1,1,1,1,2]=>3
[2,1,1,1,2,1]=>4
[2,1,1,1,3]=>3
[2,1,1,2,1,1]=>4
[2,1,1,2,2]=>3
[2,1,1,3,1]=>4
[2,1,1,4]=>3
[2,1,2,1,1,1]=>4
[2,1,2,1,2]=>5
[2,1,2,2,1]=>4
[2,1,2,3]=>4
[2,1,3,1,1]=>4
[2,1,3,2]=>4
[2,1,4,1]=>4
[2,1,5]=>3
[2,2,1,1,1,1]=>2
[2,2,1,1,2]=>3
[2,2,1,2,1]=>4
[2,2,1,3]=>3
[2,2,2,1,1]=>2
[2,2,2,2]=>1
[2,2,3,1]=>3
[2,2,4]=>2
[2,3,1,1,1]=>3
[2,3,1,2]=>4
[2,3,2,1]=>4
[2,3,3]=>2
[2,4,1,1]=>3
[2,4,2]=>3
[2,5,1]=>3
[2,6]=>2
[3,1,1,1,1,1]=>2
[3,1,1,1,2]=>3
[3,1,1,2,1]=>4
[3,1,1,3]=>3
[3,1,2,1,1]=>4
[3,1,2,2]=>3
[3,1,3,1]=>4
[3,1,4]=>3
[3,2,1,1,1]=>3
[3,2,1,2]=>4
[3,2,2,1]=>3
[3,2,3]=>3
[3,3,1,1]=>2
[3,3,2]=>2
[3,4,1]=>3
[3,5]=>2
[4,1,1,1,1]=>2
[4,1,1,2]=>3
[4,1,2,1]=>4
[4,1,3]=>3
[4,2,1,1]=>3
[4,2,2]=>2
[4,3,1]=>3
[4,4]=>1
[5,1,1,1]=>2
[5,1,2]=>3
[5,2,1]=>3
[5,3]=>2
[6,1,1]=>2
[6,2]=>2
[7,1]=>2
[8]=>1
[1,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,2]=>2
[1,1,1,1,1,1,2,1]=>3
[1,1,1,1,1,1,3]=>2
[1,1,1,1,1,2,1,1]=>3
[1,1,1,1,1,2,2]=>2
[1,1,1,1,1,3,1]=>3
[1,1,1,1,1,4]=>2
[1,1,1,1,2,1,1,1]=>3
[1,1,1,1,2,1,2]=>4
[1,1,1,1,2,2,1]=>3
[1,1,1,1,2,3]=>3
[1,1,1,1,3,1,1]=>3
[1,1,1,1,3,2]=>3
[1,1,1,1,4,1]=>3
[1,1,1,1,5]=>2
[1,1,1,2,1,1,1,1]=>3
[1,1,1,2,1,1,2]=>4
[1,1,1,2,1,2,1]=>5
[1,1,1,2,1,3]=>4
[1,1,1,2,2,1,1]=>3
[1,1,1,2,2,2]=>2
[1,1,1,2,3,1]=>4
[1,1,1,2,4]=>3
[1,1,1,3,1,1,1]=>3
[1,1,1,3,1,2]=>4
[1,1,1,3,2,1]=>4
[1,1,1,3,3]=>2
[1,1,1,4,1,1]=>3
[1,1,1,4,2]=>3
[1,1,1,5,1]=>3
[1,1,1,6]=>2
[1,1,2,1,1,1,1,1]=>3
[1,1,2,1,1,1,2]=>4
[1,1,2,1,1,2,1]=>5
[1,1,2,1,1,3]=>4
[1,1,2,1,2,1,1]=>5
[1,1,2,1,2,2]=>4
[1,1,2,1,3,1]=>5
[1,1,2,1,4]=>4
[1,1,2,2,1,1,1]=>3
[1,1,2,2,1,2]=>4
[1,1,2,2,2,1]=>3
[1,1,2,2,3]=>3
[1,1,2,3,1,1]=>4
[1,1,2,3,2]=>4
[1,1,2,4,1]=>4
[1,1,2,5]=>3
[1,1,3,1,1,1,1]=>3
[1,1,3,1,1,2]=>4
[1,1,3,1,2,1]=>5
[1,1,3,1,3]=>4
[1,1,3,2,1,1]=>4
[1,1,3,2,2]=>3
[1,1,3,3,1]=>3
[1,1,3,4]=>3
[1,1,4,1,1,1]=>3
[1,1,4,1,2]=>4
[1,1,4,2,1]=>4
[1,1,4,3]=>3
[1,1,5,1,1]=>3
[1,1,5,2]=>3
[1,1,6,1]=>3
[1,1,7]=>2
[1,2,1,1,1,1,1,1]=>3
[1,2,1,1,1,1,2]=>4
[1,2,1,1,1,2,1]=>5
[1,2,1,1,1,3]=>4
[1,2,1,1,2,1,1]=>5
[1,2,1,1,2,2]=>4
[1,2,1,1,3,1]=>5
[1,2,1,1,4]=>4
[1,2,1,2,1,1,1]=>5
[1,2,1,2,1,2]=>6
[1,2,1,2,2,1]=>5
[1,2,1,2,3]=>5
[1,2,1,3,1,1]=>5
[1,2,1,3,2]=>5
[1,2,1,4,1]=>5
[1,2,1,5]=>4
[1,2,2,1,1,1,1]=>3
[1,2,2,1,1,2]=>4
[1,2,2,1,2,1]=>5
[1,2,2,1,3]=>4
[1,2,2,2,1,1]=>3
[1,2,2,2,2]=>2
[1,2,2,3,1]=>4
[1,2,2,4]=>3
[1,2,3,1,1,1]=>4
[1,2,3,1,2]=>5
[1,2,3,2,1]=>5
[1,2,3,3]=>3
[1,2,4,1,1]=>4
[1,2,4,2]=>4
[1,2,5,1]=>4
[1,2,6]=>3
[1,3,1,1,1,1,1]=>3
[1,3,1,1,1,2]=>4
[1,3,1,1,2,1]=>5
[1,3,1,1,3]=>4
[1,3,1,2,1,1]=>5
[1,3,1,2,2]=>4
[1,3,1,3,1]=>5
[1,3,1,4]=>4
[1,3,2,1,1,1]=>4
[1,3,2,1,2]=>5
[1,3,2,2,1]=>4
[1,3,2,3]=>4
[1,3,3,1,1]=>3
[1,3,3,2]=>3
[1,3,4,1]=>4
[1,3,5]=>3
[1,4,1,1,1,1]=>3
[1,4,1,1,2]=>4
[1,4,1,2,1]=>5
[1,4,1,3]=>4
[1,4,2,1,1]=>4
[1,4,2,2]=>3
[1,4,3,1]=>4
[1,4,4]=>2
[1,5,1,1,1]=>3
[1,5,1,2]=>4
[1,5,2,1]=>4
[1,5,3]=>3
[1,6,1,1]=>3
[1,6,2]=>3
[1,7,1]=>3
[1,8]=>2
[2,1,1,1,1,1,1,1]=>2
[2,1,1,1,1,1,2]=>3
[2,1,1,1,1,2,1]=>4
[2,1,1,1,1,3]=>3
[2,1,1,1,2,1,1]=>4
[2,1,1,1,2,2]=>3
[2,1,1,1,3,1]=>4
[2,1,1,1,4]=>3
[2,1,1,2,1,1,1]=>4
[2,1,1,2,1,2]=>5
[2,1,1,2,2,1]=>4
[2,1,1,2,3]=>4
[2,1,1,3,1,1]=>4
[2,1,1,3,2]=>4
[2,1,1,4,1]=>4
[2,1,1,5]=>3
[2,1,2,1,1,1,1]=>4
[2,1,2,1,1,2]=>5
[2,1,2,1,2,1]=>6
[2,1,2,1,3]=>5
[2,1,2,2,1,1]=>4
[2,1,2,2,2]=>3
[2,1,2,3,1]=>5
[2,1,2,4]=>4
[2,1,3,1,1,1]=>4
[2,1,3,1,2]=>5
[2,1,3,2,1]=>5
[2,1,3,3]=>3
[2,1,4,1,1]=>4
[2,1,4,2]=>4
[2,1,5,1]=>4
[2,1,6]=>3
[2,2,1,1,1,1,1]=>2
[2,2,1,1,1,2]=>3
[2,2,1,1,2,1]=>4
[2,2,1,1,3]=>3
[2,2,1,2,1,1]=>4
[2,2,1,2,2]=>3
[2,2,1,3,1]=>4
[2,2,1,4]=>3
[2,2,2,1,1,1]=>2
[2,2,2,1,2]=>3
[2,2,2,2,1]=>2
[2,2,2,3]=>2
[2,2,3,1,1]=>3
[2,2,3,2]=>3
[2,2,4,1]=>3
[2,2,5]=>2
[2,3,1,1,1,1]=>3
[2,3,1,1,2]=>4
[2,3,1,2,1]=>5
[2,3,1,3]=>4
[2,3,2,1,1]=>4
[2,3,2,2]=>3
[2,3,3,1]=>3
[2,3,4]=>3
[2,4,1,1,1]=>3
[2,4,1,2]=>4
[2,4,2,1]=>4
[2,4,3]=>3
[2,5,1,1]=>3
[2,5,2]=>3
[2,6,1]=>3
[2,7]=>2
[3,1,1,1,1,1,1]=>2
[3,1,1,1,1,2]=>3
[3,1,1,1,2,1]=>4
[3,1,1,1,3]=>3
[3,1,1,2,1,1]=>4
[3,1,1,2,2]=>3
[3,1,1,3,1]=>4
[3,1,1,4]=>3
[3,1,2,1,1,1]=>4
[3,1,2,1,2]=>5
[3,1,2,2,1]=>4
[3,1,2,3]=>4
[3,1,3,1,1]=>4
[3,1,3,2]=>4
[3,1,4,1]=>4
[3,1,5]=>3
[3,2,1,1,1,1]=>3
[3,2,1,1,2]=>4
[3,2,1,2,1]=>5
[3,2,1,3]=>4
[3,2,2,1,1]=>3
[3,2,2,2]=>2
[3,2,3,1]=>4
[3,2,4]=>3
[3,3,1,1,1]=>2
[3,3,1,2]=>3
[3,3,2,1]=>3
[3,3,3]=>1
[3,4,1,1]=>3
[3,4,2]=>3
[3,5,1]=>3
[3,6]=>2
[4,1,1,1,1,1]=>2
[4,1,1,1,2]=>3
[4,1,1,2,1]=>4
[4,1,1,3]=>3
[4,1,2,1,1]=>4
[4,1,2,2]=>3
[4,1,3,1]=>4
[4,1,4]=>3
[4,2,1,1,1]=>3
[4,2,1,2]=>4
[4,2,2,1]=>3
[4,2,3]=>3
[4,3,1,1]=>3
[4,3,2]=>3
[4,4,1]=>2
[4,5]=>2
[5,1,1,1,1]=>2
[5,1,1,2]=>3
[5,1,2,1]=>4
[5,1,3]=>3
[5,2,1,1]=>3
[5,2,2]=>2
[5,3,1]=>3
[5,4]=>2
[6,1,1,1]=>2
[6,1,2]=>3
[6,2,1]=>3
[6,3]=>2
[7,1,1]=>2
[7,2]=>2
[8,1]=>2
[9]=>1
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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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