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A329746 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.

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