OFFSET
1,8
COMMENTS
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.
EXAMPLE
Triangle begins:
1
1 1
1 1 1
1 2 1 1
1 1 2 3 0
1 3 4 3 0 0
1 1 4 8 1 0 0
1 3 6 10 2 0 0 0
1 2 8 13 6 0 0 0 0
1 3 11 20 7 0 0 0 0 0
1 1 11 29 14 0 0 0 0 0 0
1 5 19 31 20 1 0 0 0 0 0 0
1 1 17 50 30 2 0 0 0 0 0 0 0
1 3 25 64 37 5 0 0 0 0 0 0 0 0
1 3 29 74 62 7 0 0 0 0 0 0 0 0 0
Row n = 8 counts the following partitions:
(8) (44) (53) (332) (4211)
(2222) (62) (422) (32111)
(11111111) (71) (611)
(431) (3221)
(521) (5111)
(3311) (22211)
(41111)
(221111)
(311111)
(2111111)
MATHEMATICA
runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;
Table[Length[Select[IntegerPartitions[n], runsres[#]==k&]], {n, 10}, {k, 0, n-1}]
PROG
(PARI) \\ rr(p) gives runs resistance of partition.
rr(p)={my(r=0); while(#p > 1, my(L=List(), k=0); for(i=1, #p, if(i==#p||p[i]<>p[i+1], listput(L, i-k); k=i)); p=Vec(L); r++); r}
row(n)={my(v=vector(n)); forpart(p=n, v[1+rr(Vec(p))]++); v}
{ for(n=1, 10, print(row(n))) } \\ Andrew Howroyd, Jan 19 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Nov 21 2019
STATUS
approved