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A125106
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Enumeration of partitions by binary representation: each 1 is a part; the part size is 1 more than the number of zeros in the rest of the number.
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11
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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, 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, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Another way to describe this: starting with the binary representation and a counter set at one, count the zeros from right to left. Write a term equal to the counter for each "1" encountered.
A101211 is a similar sequence with A005811 elements per row which maps natural numbers to compositions (ordered partitions).
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.
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.
Taking each partition as a separate row, the row lengths are A000120, and the row sums are A161511.
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FORMULA
| 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.
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EXAMPLE
| Row 4:
1000 [4]
1001 [3,1]
1010 [3,2]
1011 [2,1,1]
1100 [3,3]
1101 [2,2,1]
1110 [2,2,2]
1111 [1,1,1,1]
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MATHEMATICA
| 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}]] (* From Jean-François Alcover, Jan 24 2012 *)
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CROSSREFS
| Cf. A000041,A000120,A161511,A005811,A101211,A001792,A049611,A126411.
Cf. A005940. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Mar 06 2010]
Sequence in context: A192710 A112380 A165162 * A152538 A141110 A190770
Adjacent sequences: A125103 A125104 A125105 * A125107 A125108 A125109
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KEYWORD
| tabf,nice,nonn
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AUTHOR
| Alford Arnold (Alford1940(AT)aol.com), Dec 10 2006
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EXTENSIONS
| Edited by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 11 2009
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