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EXAMPLE
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a(0) = 1;
a(1) = C(0,0)*a(0)*1!/0! = 1;
a(2) = C(1,1)*a(1)*2!/1! + C(1,0)*a(0)*2!/0! = 4;
a(3) = C(2,2)*a(2)*3!/2! + C(2,1)*a(1)*3!/1! + C(2,0)*a(0)*3!/0! = 30;
a(4) = C(3,3)*a(3)*4!/3! + C(3,2)*a(2)*4!/2! + C(3,1)*a(1)*4!/1! + C(3,0)*a(0)*4!/0! = 360.
Illustration as family of lists of sublists extending set partitions.
In this interpretation the lowercase letters allow us to distinguish between integers introduced at each iteration (or generation).
Construction from the family of size n to family of size n+1 is done by insertion.
Insertion is only possible at the end of a sublist or to create a new singleton sublist at the end of the list.
:
1: {{1a}}*
4: {{1a},{1b}} {{1a,1b}} {{1a,2b}}* {{1a},{2b}}*
30: {{1a,1c},{1b}} {{1a},{1b,1c}} {{1a},{1b},{1c}}
....{{1a,2c},{1b}} {{1a},{1b,2c}} {{1a},{1b},{2c}}
....{{1a,3c},{1b}} {{1a},{1b,3c}} {{1a},{1b},{3c}}
....{{1a,1b,1c}} {{1a,1b},{1c}}
....{{1a,1b,2c}} {{1a,1b},{2c}}
....{{1a,1b,3c}} {{1a,1b},{3c}}
....{{1a,2b,1c}} {{1a,2b,2c}} {{1a,2b,3c}}*
....{{1a,2b},{1c}} {{1a,2b},{2c}} {{1a,2b},{3c}}*
....{{1a,1c},{2b}} {{1a},{2b,1c}} {{1a},{2b},{1c}}
....{{1a,2c},{2b}} {{1a},{2b,2c}} {{1a},{2b},{2c}}
....{{1a,3c},{2b}}* {{1a},{2b,3c}}* {{1a},{2b},{3c}}*
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The lists of sublists marked with * correspond to classical set partitions counted by Bell numbers A000110. (End)
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