A210941 -id:A210941 - cp's OEIS frontend

A141285 Largest part of the n-th partition of j in the list of colexicographically ordered partitions of j, if 1 <= n <= A000041(j).

1, 2, 3, 2, 4, 3, 5, 2, 4, 3, 6, 3, 5, 4, 7, 2, 4, 3, 6, 5, 4, 8, 3, 5, 4, 7, 3, 6, 5, 9, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 6, 5, 9, 4, 8, 7, 6, 12

Offset: 1

Omar E. Pol, Aug 01 2008

Also largest part of the n-th region of the set of partitions of j, if 1 <= n <= A000041(j). For the definition of "region of the set of partitions of j" see A206437.
Also triangle read by rows: T(j,k) is the largest part of the k-th region in the last section of the set of partitions of j.
For row n >= 2 the rows of triangle are also the branches of a tree which is a projection of a three-dimensional structure of the section model of partitions of A135010, version tree. The branches of even rows give A182730. The branches of odd rows give A182731. Note that each column contains parts of the same size. It appears that the structure of A135010 is a periodic table of integer partitions. See also A210979 and A210980.
Also column 1 of: A193870, A206437, A210941, A210942, A210943. - Omar E. Pol, Sep 01 2013
Also row lengths of A211009. - Omar E. Pol, Feb 06 2014

			Written as a triangle T(j,k) the sequence begins:
  1;
  2;
  3;
  2, 4;
  3, 5;
  2, 4, 3, 6;
  3, 5, 4, 7;
  2, 4, 3, 6, 5, 4, 8;
  3, 5, 4, 7, 3, 6, 5, 9;
  2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10;
  3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8,  7, 6, 11;
  ...
  ------------------------------------------
  n  A000041                a(n)
  ------------------------------------------
   1 = p(1)                   1
   2 = p(2)                 2 .
   3 = p(3)                   . 3
   4                        2 .
   5 = p(4)               4   .
   6                          . 3
   7 = p(5)                   .   5
   8                        2 .
   9                      4   .
  10                    3     .
  11 = p(6)           6       .
  12                          . 3
  13                          .   5
  14                          .     4
  15 = p(7)                   .       7
  ...
From _Omar E. Pol_, Aug 22 2013: (Start)
Illustration of initial terms (n = 1..11) in three ways: as the largest parts of the partitions of 6 (see A026792), also as the largest parts of the regions of the diagram, also as the diagonal of triangle. By definition of "region" the largest part of the n-th region is also the largest part of the n-th partition (see below):
  --------------------------------------------------------
  .                  Diagram         Triangle in which
  Partitions       of regions       rows are partitions
  of 6           and partitions   and columns are regions
  --------------------------------------------------------
  .                _ _ _ _ _ _
  6                _ _ _      |                         6
  3+3              _ _ _|_    |                       3 3
  4+2              _ _    |   |                     4   2
  2+2+2            _ _|_ _|_  |                   2 2   2
  5+1              _ _ _    | |                 5       1
  3+2+1            _ _ _|_  | |               3 1       1
  4+1+1            _ _    | | |             4   1       1
  2+2+1+1          _ _|_  | | |           2 2   1       1
  3+1+1+1          _ _  | | | |         3   1   1       1
  2+1+1+1+1        _  | | | | |       2 1   1   1       1
  1+1+1+1+1+1       | | | | | |     1 1 1   1   1       1
  ...
The equivalent sequence for compositions is A001511. Explanation: for the positive integer j the diagram of regions of the set of compositions of j has 2^(j-1) regions. The largest part of the n-th region is A001511(n). The number of parts is A006519(n). On the other hand the diagram of regions of the set of partitions of j has A000041(j) regions. The largest part of the n-th region is a(n) = A001511(A228354(n)). The number of parts is A194446(n). Both diagrams have j sections. The diagram for partitions can be interpreted as one of the three views of a three dimensional diagram of compositions in which the rows of partitions are in orthogonal direction to the rest. For the first five sections of the diagrams see below:
  --------------------------------------------------------
  .          Diagram                           Diagram
  .         of regions                        of regions
  .      and compositions                   and partitions
  ---------------------------------------------------------
  .      j = 1 2 3 4 5                     j = 1 2 3 4 5
  ---------------------------------------------------------
   n  A001511                    A228354  a(n)
  ---------------------------------------------------------
   1   1     _| | | | | ............ 1    1    _| | | | |
   2   2     _ _| | | | ............ 2    2    _ _| | | |
   3   1     _|   | | |    ......... 4    3    _ _ _| | |
   4   3     _ _ _| | | ../  ....... 6    2    _ _|   | |
   5   1     _| |   | |    / ....... 8    4    _ _ _ _| |
   6   2     _ _|   | | ../ /   .... 12   3    _ _ _|   |
   7   1     _|     | |    /   /   . 16   5    _ _ _ _ _|
   8   4     _ _ _ _| | ../   /   /
   9   1     _| | |   |      /   /
  10   2     _ _| |   |     /   /
  11   1     _|   |   |    /   /
  12   3     _ _ _|   | ../   /
  13   1     _| |     |      /
  14   2     _ _|     |     /
  15   1     _|       |    /
  16   5     _ _ _ _ _| ../
  ...
Also we can draw an infinite Dyck path in which the n-th odd-indexed line segment has a(n) up-steps and the n-th even-indexed line segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two successive valleys at height 0 is also the partition number A000041(n). See below:
.                                 5
.                                 /\                 3
.                   4            /  \           4    /\
.                   /\          /    \          /\  /
.         3        /  \     3  /      \        /  \/
.    2    /\   2  /    \    /\/        \   2  /
. 1  /\  /  \  /\/      \  /            \  /\/
. /\/  \/    \/          \/              \/
.
.(End)

Alois P. Heinz, Table of n, a(n) for n = 1..4000
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of the seven regions of 5

Where records occur give A000041, n>=1. Column 1 is A158478. Row j has length A187219(j). Row sums give A138137. Right border gives A000027.

Cf. A000041, A135010, A182730, A182731, A182732, A182733, A182982, A182983, A182703, A193870, A194446, A194447, A194550, A206437, A210979, A210980, A211978, A220517, A225600, A225610.

Last/@DeleteCases[DeleteCases[Sort@PadRight[Reverse/@IntegerPartitions[13]], x_ /; x == 0, 2], {}] (* updated _Robert Price, May 15 2020 *)

a(n) = A001511(A228354(n)). - Omar E. Pol, Aug 22 2013

Edited by Omar E. Pol, Nov 28 2010

Better definition and edited by Omar E. Pol, Oct 17 2013

A194602 Integer partitions interpreted as binary numbers.

Original entry on oeis.org

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

Offset: 0

Tilman Piesk, Aug 30 2011

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)
Numbers whose binary representation has runs of '1's of weakly increasing length (with trailing '0's (introducing a run of length 0) forbidden, i.e., only odd terms beyond 0). - M. F. Hasler, May 14 2020

			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)

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
Index entries for sequences related to binary expansion of n

Cf. A000041 (partition numbers).
Cf. A002865 (row lengths).
Cf. A002450, A000225 (subsequences).
Cf. A249544 (rows and columns are subsequences).
Cf. A210941, A194548, A141285, A228354, A326956.

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 *)

isA194602(n) = if(!n,1,if(!(n%2),0,my(prl=0,rl=0); while(n, if(0==(n%2),if((prl && rl>prl)||0==(n%4), return(0)); prl=rl; rl=0, rl++); n >>= 1); ((0==prl)||(rl<=prl)))); \\ - Antti Karttunen, Dec 06 2021

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
a(n) = A228354(1+n) - 1. - Antti Karttunen, Dec 06 2021

Comments edited by Li-yao Xia, May 13 2014

Incorrect PARI-program removed by Antti Karttunen, Dec 09 2021

A210942 Triangle read by rows in which row n lists the parts > 1 of the n-th region of the shell model of partitions, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 3, 5, 2, 2, 4, 2, 3, 6, 3, 2, 2, 3, 5, 2, 4, 7, 3, 2, 2, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 3, 5, 2, 4, 7, 3, 2, 2, 3, 6, 3, 5, 9, 4, 3, 3, 2, 2, 2, 2, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 4, 7, 3, 6, 5, 10, 5

Offset: 1

Omar E. Pol, Apr 18 2012

For the definition of "region of n" see A206437. See also A186114. Row n lists the largest part and the parts > 1 of the n-th region of the shell model of partitions. Also 1 together with the numbers > 1 of A206437.

			Written as a triangle begins:
1;
2;
3;
2;
4,2;
3;
5,2,
2;
4,2;
3;
6,3,2,2;
3;
5,2;
4;
7,3,2,2;

Omar E. Pol, Illustration of the seven regions of 5
Omar E. Pol, Illustration of the partitions and regions of n, for n = 1..12

Column 1 is A141285. Records give A000027. The n-th record is T(A000041(n),1).

Cf. A135010, A138121, A182699, A182709, A183152, A186114, A187219, A194436-A194439, A194447-A194448, A196025, A198381, A206437, A210941.

A210943 Square array read by antidiagonals in which row n lists the parts of the infinite n-th zone of the shell model of partitions, in nonincreasing order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 19 2012

The n-th zone of the shell model of partitions is formed by the parts of the n-th row of triangle A210941 together with infinitely many parts of size 1.

			Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
2, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
3, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
2, 2, 1, 1, 1, 1, 1, 1, 1, 1,...
4, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
3, 2, 1, 1, 1, 1, 1, 1, 1, 1,...
5, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
2, 2, 2, 1, 1, 1, 1, 1, 1, 1,...
4, 2, 1, 1, 1, 1, 1, 1, 1, 1,...
3, 3, 1, 1, 1, 1, 1, 1, 1, 1,...
6, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
3, 2, 2, 1, 1, 1, 1, 1, 1, 1,...
5, 2, 1, 1, 1, 1, 1, 1, 1, 1,...
4, 3, 1, 1, 1, 1, 1, 1, 1, 1,...
7, 1, 1, 1, 1, 1, 1, 1, 1, 1,...

Omar E. Pol, Illustration of the partitions and regions of n, for n = 1..12

Column 1 is A141285.

Cf. A135010, A138121, A182742, A182743, A210941, A210942.

cp's OEIS Frontend

A141285 Largest part of the n-th partition of j in the list of colexicographically ordered partitions of j, if 1 <= n <= A000041(j).

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A194602 Integer partitions interpreted as binary numbers.

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A210942 Triangle read by rows in which row n lists the parts > 1 of the n-th region of the shell model of partitions, with a(1) = 1.

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A210943 Square array read by antidiagonals in which row n lists the parts of the infinite n-th zone of the shell model of partitions, in nonincreasing order.

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