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%I #10 Jan 19 2023 22:35:17
%S 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,
%T 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,
%U 0,1,5,19,31,20,1,0,0,0,0,0,0
%N Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 with runs-resistance k, 0 <= k <= n - 1.
%C 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.
%H Andrew Howroyd, <a href="/A329746/b329746.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%e Triangle begins:
%e 1
%e 1 1
%e 1 1 1
%e 1 2 1 1
%e 1 1 2 3 0
%e 1 3 4 3 0 0
%e 1 1 4 8 1 0 0
%e 1 3 6 10 2 0 0 0
%e 1 2 8 13 6 0 0 0 0
%e 1 3 11 20 7 0 0 0 0 0
%e 1 1 11 29 14 0 0 0 0 0 0
%e 1 5 19 31 20 1 0 0 0 0 0 0
%e 1 1 17 50 30 2 0 0 0 0 0 0 0
%e 1 3 25 64 37 5 0 0 0 0 0 0 0 0
%e 1 3 29 74 62 7 0 0 0 0 0 0 0 0 0
%e Row n = 8 counts the following partitions:
%e (8) (44) (53) (332) (4211)
%e (2222) (62) (422) (32111)
%e (11111111) (71) (611)
%e (431) (3221)
%e (521) (5111)
%e (3311) (22211)
%e (41111)
%e (221111)
%e (311111)
%e (2111111)
%t runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t Table[Length[Select[IntegerPartitions[n],runsres[#]==k&]],{n,10},{k,0,n-1}]
%o (PARI) \\ rr(p) gives runs resistance of partition.
%o 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}
%o row(n)={my(v=vector(n)); forpart(p=n, v[1+rr(Vec(p))]++); v}
%o { for(n=1, 10, print(row(n))) } \\ _Andrew Howroyd_, Jan 19 2023
%Y Row sums are A000041.
%Y Column k = 1 is A032741.
%Y Column k = 2 is A329745.
%Y A similar invariant is frequency depth; see A323014, A325280.
%Y The version for compositions is A329744.
%Y The version for binary words is A329767.
%Y Cf. A098504, A182850, A225485, A242882, A318928, A325410, A329747.
%K nonn,tabl
%O 1,8
%A _Gus Wiseman_, Nov 21 2019
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