Identifier
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
click to show generating function       
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].
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