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%I #45 Jan 17 2023 21:24:51
%S 1,2,1,1,3,2,1,2,2,1,1,1,4,3,1,3,2,2,1,1,3,3,2,2,1,2,2,2,1,1,1,1,5,4,
%T 1,4,2,3,1,1,4,3,3,2,1,3,2,2,2,1,1,1,4,4,3,3,1,3,3,2,2,2,1,1,3,3,3,2,
%U 2,2,1,2,2,2,2,1,1,1,1,1
%N Enumeration of partitions by binary representation: each 1 is a part; the part size is 1 more than the number of 0's in the rest of the number.
%C Another way to describe this: starting with the binary representation and a counter set at one, count the 0's from right to left. Write a term equal to the counter for each "1" encountered.
%C A101211 is a similar sequence, with A005811 elements per row which maps natural numbers to compositions (ordered partitions).
%C There are two ways to consider this as a table: taking each partition as a row, or taking the partitions generated by 2^(n-1) through 2^n-1 as a row.
%C Taking the n-th row as multiple partitions, it consists of those partitions with the first hook size (largest part plus number of parts minus 1) equal to n. The number of integers in this n-th row is A001792(n-1), and the row sum is A049611.
%C Taking each partition as a separate row, the row lengths are A000120, and the row sums are A161511.
%C Heinz numbers of the rows are A005940. - _Gus Wiseman_, Jan 17 2023
%H Alois P. Heinz, <a href="/A125106/b125106.txt">Rows n = 1..12, flattened</a>
%F Partition 2n is partition n with every part size increased by 1; partition 2n+1 is partition n with an additional part of size 1.
%F T(n,k) = A272020(n,k) - A000120(n) + k. - _Gus Wiseman_, Jan 17 2023
%e Row 4:
%e 1000 [4]
%e 1001 [3,1]
%e 1010 [3,2]
%e 1011 [2,1,1]
%e 1100 [3,3]
%e 1101 [2,2,1]
%e 1110 [2,2,2]
%e 1111 [1,1,1,1]
%p b:= proc(n) local c, l, m; l:=[][]; m:= n; c:=1;
%p while m>0 do if irem(m, 2, 'm')=0 then c:= c+1
%p else l:= c, l fi
%p od; l
%p end:
%p T:= n-> seq(b(i), i=2^(n-1)..2^n-1):
%p seq(T(n), n=1..7); # _Alois P. Heinz_, Sep 25 2015
%t f[k_] := (bits = IntegerDigits[k, 2]; zerosCount = Reverse[ Accumulate[ 1-Reverse[bits] ] ] + 1; Select[ Transpose[ {bits, zerosCount} ], First[#] == 1 & ][[All, 2]]); row[n_] := Table[ f[k], {k, 2^(n-1), 2^n-1}]; Flatten[ Table[ row[n], {n, 1, 5}]] (* _Jean-François Alcover_, Jan 24 2012 *)
%t scc[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Table[Reverse[scc[n]-Range[Length[scc[n]]]+1],{n,0,20}] (* _Gus Wiseman_, Jan 17 2023 *)
%Y Cf. A000041, A005811, A037016, A101211, A001792, A049611, A126411.
%Y Each partition as row: A000120 (row widths), A161511 (row sums), A243499 (row products).
%Y Cf. A005940. - _Franklin T. Adams-Watters_, Mar 06 2010
%Y Lasts are A001511.
%Y Firsts are A008687.
%Y Cf. A019565, A029837, A029931, A048793, A059893, A066099, A070939, A242628.
%K tabf,nice,nonn
%O 1,2
%A _Alford Arnold_, Dec 10 2006
%E Edited by _Franklin T. Adams-Watters_, Jun 11 2009
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