A096111 -id:A096111 - cp's OEIS frontend

A358333 By concatenating the standard compositions for each part of the n-th standard composition, we get a sequence of length a(n). Row-lengths of A357135.

0, 1, 1, 2, 2, 2, 2, 3, 1, 3, 2, 3, 3, 3, 3, 4, 2, 2, 3, 4, 3, 3, 3, 4, 2, 4, 3, 4, 4, 4, 4, 5, 2, 3, 2, 3, 4, 4, 4, 5, 2, 4, 3, 4, 4, 4, 4, 5, 3, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 5, 5, 5, 6, 3, 3, 3, 4, 3, 3, 3, 4, 3, 5, 4, 5, 5, 5, 5, 6, 3, 3, 4, 5, 4, 4, 4

Offset: 0

Gus Wiseman, Nov 10 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			Composition 92 in standard order is (2,1,1,3), with compositions ((2),(1),(1),(1,1)) so a(92) = 5.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions (A066099).
Dominates A000120.
Row-lengths of A357135, which is ranked by A357134.
A related sequence is A358330.
Cf. A001511, A029931, A048896, A058891, A070939, A096111, A333766, A357137, A357139, A357186, A357187.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Length/@Table[Join@@stc/@stc[n],{n,0,100}]

Sum of A000120 over row n of A066099.

A371732 Numbers n such that each binary index k (from row n of A048793) has the same sum of binary indices A029931(k).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 32, 64, 128, 144, 256, 288, 512, 576, 1024, 2048, 3072, 4096, 8192, 16384, 32768, 32800, 33024, 33056, 65536, 65600, 66048, 66112, 131072, 132096, 133120, 134144, 262144, 266240, 524288, 528384, 786432, 790528, 1048576, 1056768, 2097152

Offset: 1

Gus Wiseman, Apr 13 2024

			The terms together with their binary expansions and binary indices begin:
        1:                1 ~ {1}
        2:               10 ~ {2}
        4:              100 ~ {3}
        8:             1000 ~ {4}
       12:             1100 ~ {3,4}
       16:            10000 ~ {5}
       32:           100000 ~ {6}
       64:          1000000 ~ {7}
      128:         10000000 ~ {8}
      144:         10010000 ~ {5,8}
      256:        100000000 ~ {9}
      288:        100100000 ~ {6,9}
      512:       1000000000 ~ {10}
      576:       1001000000 ~ {7,10}
     1024:      10000000000 ~ {11}
     2048:     100000000000 ~ {12}
     3072:     110000000000 ~ {11,12}
     4096:    1000000000000 ~ {13}
     8192:   10000000000000 ~ {14}
    16384:  100000000000000 ~ {15}
    32768: 1000000000000000 ~ {16}
    32800: 1000000000100000 ~ {6,16}

For prime instead of binary indices we have A326534.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A321142 and A371794 count non-biquanimous strict partitions.
A321452 counts quanimous partitions, ranks A321454.
A326031 gives weight of the set-system with BII-number n.
A357976 ranks the biquanimous partitions counted by A002219 aerated.
A371731 ranks the non-biquanimous partitions counted by A371795, A006827.
Cf. A035470, A038041, A237258, A320324, A321453, A321455, A326518, A336137, A371783, A371791, A371796.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],SameQ@@Total/@bix/@bix[#]&]

A373120 Number of distinct possible binary ranks of integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 33, 43, 55, 70, 89, 109, 136, 167, 206, 251, 306, 371, 445, 535, 639, 759, 904, 1069, 1262, 1489, 1747, 2047, 2390, 2784, 3237, 3754, 4350, 5027, 5798, 6680, 7671, 8808, 10091, 11543, 13190, 15040, 17128, 19477, 22118

Offset: 0

Gus Wiseman, May 26 2024

nonn

			The partitions of 4 are (4), (3,1), (2,2), (2,1,1), (1,1,1,1), with respective binary ranks 8, 5, 4, 4, 4, so a(4) = 3.

The strict case is A000009.
A048675 gives binary rank of prime indices, distinct A087207.
A118462 lists binary ranks of strict integer partitions, row sums A372888.
A277905 groups all positive integers by binary rank of prime indices.
A372890 adds up binary ranks of integer partitions.
Binary indices (A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- max A029837 or A070939, opposite A070940
- sum A029931, product A096111
- reverse A272020
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000041, A018819, A158704, A158705, A225620, A231204, A372688.

Table[Length[Union[Total[2^(#-1)]&/@IntegerPartitions[n]]],{n,0,15}]

cp's OEIS Frontend

A358333 By concatenating the standard compositions for each part of the n-th standard composition, we get a sequence of length a(n). Row-lengths of A357135.

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A371732 Numbers n such that each binary index k (from row n of A048793) has the same sum of binary indices A029931(k).

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A373120 Number of distinct possible binary ranks of integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

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Author

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Examples

Crossrefs

Programs

Mathematica