login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A194602 Integer partitions interpreted as binary numbers. 19
0, 1, 3, 5, 7, 11, 15, 21, 23, 27, 31, 43, 47, 55, 63, 85, 87, 91, 95, 111, 119, 127, 171, 175, 183, 191, 219, 223, 239, 255, 341, 343, 347, 351, 367, 375, 383, 439, 447, 479, 495, 511, 683, 687, 695, 703, 731, 735, 751, 767, 879, 887, 895, 959, 991, 1023, 1365, 1367, 1371, 1375, 1391 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The 2^(n-1) compositions of n correspond to binary numbers, and the partitions of n can be seen as compositions with addends ordered by size, so they also correspond to binary numbers.

The finite sequence for partitions of n (ordered by size) is the beginning of the sequence for partitions of n+1, which leads to an infinite sequence.

From Tilman Piesk, Jan 30 2016: (Start)

It makes sense to regard the positive values as a triangle with row lengths A002865(n) and row numbers n>=2. In this triangle row n contains all partitions of n with non-one addends only. See link "Triangle with Young diagrams".

This sequence contains all binary palindromes with m runs of n ones separated by single zeros. They are ordered in the array A249544. All the rows and columns of this array are subsequences of this sequence, notably its top row (A000225, the powers of two minus one).

Sequences by Omar E. Pol: The "triangle" A210941 defines the same sequence of partitions. Its n-th row shows the non-one addends of the n-th partition. There are A194548(n) of them, and A141285(n) is the largest among them. (The "triangle" A210941 does not actually form a triangle, but A210941 and A141285 do.) Note that the offset of these sequences is 1 and not 0.

(End)

LINKS

Tilman Piesk, Table of n, a(n) for n = 0..8348

Tilman Piesk, Same table with binary strings and non-one addends

Tilman Piesk, Triangle with Young diagrams (n = 2..20).

Tilman Piesk, Integer partitions and Permutations and partitions in the OEIS

Tilman Piesk, Python functions, keynum_to_valnum(n) = a(n), valnum_to_keynum(a(n)) = n.

Li-yao Xia, Identities for A194602

FORMULA

a( A000041(n)-1 ) = A000225(n-1) for n>=1. - Tilman Piesk, Apr 16 2012

a( A000041(2n-1) ) = A002450(n)  for n>=1. - Tilman Piesk, Apr 16 2012

a( A249543 ) = A249544. - Tilman Piesk, Oct 31 2014

EXAMPLE

From Joerg Arndt, Nov 17 2012: (Start)

With leading zeros included, the first A000041(n) terms correspond to the list of partitions of n as nondecreasing compositions in lexicographic order.

For example, the first A000041(10)=42 terms correspond to the partitions of 10 as follows (dots for zeros in the binary expansions):

[ n]   binary(a(n))  a(n)  partition

[ 0]   ..........     0    [ 1 1 1 1 1 1 1 1 1 1 ]

[ 1]   .........1     1    [ 1 1 1 1 1 1 1 1 2 ]

[ 2]   ........11     3    [ 1 1 1 1 1 1 1 3 ]

[ 3]   .......1.1     5    [ 1 1 1 1 1 1 2 2 ]

[ 4]   .......111     7    [ 1 1 1 1 1 1 4 ]

[ 5]   ......1.11    11    [ 1 1 1 1 1 2 3 ]

[ 6]   ......1111    15    [ 1 1 1 1 1 5 ]

[ 7]   .....1.1.1    21    [ 1 1 1 1 2 2 2 ]

[ 8]   .....1.111    23    [ 1 1 1 1 2 4 ]

[ 9]   .....11.11    27    [ 1 1 1 1 3 3 ]

[10]   .....11111    31    [ 1 1 1 1 6 ]

[11]   ....1.1.11    43    [ 1 1 1 2 2 3 ]

[12]   ....1.1111    47    [ 1 1 1 2 5 ]

[13]   ....11.111    55    [ 1 1 1 3 4 ]

[14]   ....111111    63    [ 1 1 1 7 ]

[15]   ...1.1.1.1    85    [ 1 1 2 2 2 2 ]

[16]   ...1.1.111    87    [ 1 1 2 2 4 ]

[17]   ...1.11.11    91    [ 1 1 2 3 3 ]

[18]   ...1.11111    95    [ 1 1 2 6 ]

[19]   ...11.1111   111    [ 1 1 3 5 ]

[20]   ...111.111   119    [ 1 1 4 4 ]

[21]   ...1111111   127    [ 1 1 8 ]

[22]   ..1.1.1.11   171    [ 1 2 2 2 3 ]

[23]   ..1.1.1111   175    [ 1 2 2 5 ]

[24]   ..1.11.111   183    [ 1 2 3 4 ]

[25]   ..1.111111   191    [ 1 2 7 ]

[26]   ..11.11.11   219    [ 1 3 3 3 ]

[27]   ..11.11111   223    [ 1 3 6 ]

[28]   ..111.1111   239    [ 1 4 5 ]

[29]   ..11111111   255    [ 1 9 ]

[30]   .1.1.1.1.1   341    [ 2 2 2 2 2 ]

[31]   .1.1.1.111   343    [ 2 2 2 4 ]

[32]   .1.1.11.11   347    [ 2 2 3 3 ]

[33]   .1.1.11111   351    [ 2 2 6 ]

[34]   .1.11.1111   367    [ 2 3 5 ]

[35]   .1.111.111   375    [ 2 4 4 ]

[36]   .1.1111111   383    [ 2 8 ]

[37]   .11.11.111   439    [ 3 3 4 ]

[38]   .11.111111   447    [ 3 7 ]

[39]   .111.11111   479    [ 4 6 ]

[40]   .1111.1111   495    [ 5 5 ]

[41]   .111111111   511    [ 10 ]

(End)

MATHEMATICA

lim = 12;

Sort[FromDigits[Reverse@ #, 2] & /@

   Map[If[Length@ # == 0, {0}, Flatten@ Most@ #] &@

     Riffle[#, Table[0, Length@ #]] &,

     Map[Table[1, # - 1] &,

       Sort@ IntegerPartitions@ lim /. 1 -> Nothing, {2}]]]

(* Michael De Vlieger, Feb 14 2016 *)

CROSSREFS

Cf. A000041 (partition numbers).

Cf. A002865 (row lengths).

Cf. A002450, A000225 (subsequences).

Cf. A249544 (rows and columns are subsequences).

Cf. A210941, A194548, A141285.

Sequence in context: A219655 A007665 A208994 * A177139 A252793 A062488

Adjacent sequences:  A194599 A194600 A194601 * A194603 A194604 A194605

KEYWORD

nonn,tabf

AUTHOR

Tilman Piesk, Aug 30 2011

EXTENSIONS

Comments edited by Li-yao Xia, May 13 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified July 27 05:30 EDT 2016. Contains 275064 sequences.