A114901 -id:A114901 - cp's OEIS frontend

A353508 Number of integer compositions of n with no ones or runs of length 1.

1, 0, 0, 0, 1, 0, 2, 0, 2, 1, 4, 0, 8, 2, 11, 4, 21, 5, 37, 12, 57, 25, 104, 38, 177, 79, 292, 149, 513, 251, 876, 482, 1478, 871, 2562, 1533, 4387, 2815, 7473, 5036, 12908, 8935, 22135, 16085, 37940, 28611, 65422, 50731, 112459, 90408, 193386, 160119, 333513

Offset: 0

Gus Wiseman, May 17 2022

nonn

			The a(0) = 1 through a(14) = 11 compositions (empty columns indicated by dots, 0 is the empty composition):
  0  .  .  .  22  .  33   .  44    333  55     .  66      22333  77
                     222     2222       2233      444     33322  2255
                                        3322      2244           3344
                                        22222     3333           4433
                                                  4422           5522
                                                  22233          22244
                                                  33222          44222
                                                  222222         222233
                                                                 223322
                                                                 332222
                                                                 2222222

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The version for partitions is A339222.
Compositions counted by their run-lengths:
- For run-lengths <= 1 we have A003242, ranked by A333489.
- For run-lengths = 2 we have A003242 aerated.
- For run-lengths > 1 we have A114901, ranked by A353427.
- For run-lengths <= 2 we have A128695 matching A335464.
- For run-lengths > 2 we have A353400, partitions A100405.
- For run-lengths all prime we have A353401.
- For run-lengths and parts > 2 we have A353428.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A106356 counts compositions by number of adjacent equal parts.
A261983 counts non-anti-run compositions.
A274174 counts compositions with equal parts contiguous.
Cf. A005811, A078012, A238279, A329739, A333755, A351013, A353390.

b:= proc(n,h) option remember; `if`(n=0, 1, add(
     `if`(i<>h, add(b(n-i*j, i), j=2..n/i), 0), i=2..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, May 17 2022

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[#,1]&&!MemberQ[Length/@Split[#],1]&]],{n,0,15}]

a(41)-a(52) from Alois P. Heinz, May 17 2022

A329869 Number of compositions of n with runs-resistance equal to cuts-resistance minus 1.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 4, 5, 11, 19, 36, 77, 138, 252, 528, 1072, 2204, 4634, 9575, 19732, 40754

Offset: 0

Gus Wiseman, Nov 23 2019

A composition of n is a finite sequence of positive integers summing to n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.
For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The a(1) = 1 through a(9) = 19 compositions:
  1   2   3   4   5   6      7       8        9
      11      22      33     11113   44       11115
                      11112  31111   11114    12222
                      21111  111211  41111    22221
                             112111  111122   51111
                                     111311   111222
                                     113111   111411
                                     211112   114111
                                     221111   211113
                                     1111121  222111
                                     1211111  311112
                                              1111131
                                              1111221
                                              1112112
                                              1121112
                                              1221111
                                              1311111
                                              2111211
                                              2112111
For example, the runs-resistance of (1221111) is 3 because we have: (1221111) -> (124) -> (111) -> (3), while the cuts-resistance is 4 because we have: (1221111) -> (2111) -> (11) -> (1) -> (), so (1221111) is counted under a(9).

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

The version for binary indices is A329866.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Cf. A003242, A098504, A114901, A242882, A318928, A319411, A319416, A319420, A319421, A329864, A329865, A329867, A329868.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]+1==degdep[#]&]],{n,0,10}]

A353501 Number of integer partitions of n with all parts and all multiplicities > 2.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 2, 3, 0, 0, 6, 2, 0, 6, 3, 2, 9, 2, 5, 11, 3, 5, 18, 6, 4, 20, 13, 8, 26, 10, 17, 37, 14, 16, 51, 23, 24, 58, 38, 32, 75, 44, 52, 100, 52, 59, 143, 75, 77, 159, 114, 112, 203, 132, 154, 266, 175

Offset: 0

Gus Wiseman, May 16 2022

nonn

			The a(n) partitions for selected n (A = 10):
  n=9:   n=12:   n=21:      n=24:       n=30:
------------------------------------------------------
  (333)  (444)   (777)      (888)       (AAA)
         (3333)  (444333)   (6666)      (66666)
                 (3333333)  (444444)    (555555)
                            (555333)    (666444)
                            (4443333)   (777333)
                            (33333333)  (6663333)
                                        (55533333)
                                        (444333333)
                                        (3333333333)

The version for only parts > 2 is A008483.
The version for only multiplicities > 2 is A100405.
The version for parts and multiplicities > 1 is A339222, ranked by A062739.
For prime parts and multiplicities we have A351982, compositions A353429.
The version for compositions is A353428 (partial A078012, A353400).
These partitions are ranked by A353502.
A000726 counts partitions with all mults <= 2, compositions A128695.
A004250 counts partitions with some part > 2, compositions A008466.
A137200 counts compositions with all parts and run-lengths <= 2.
Cf. A003242, A047966, A055923, A098859, A114901, A325534, A333755, A335464.

Table[Length[Select[IntegerPartitions[n],Min@@#>2&&Min@@Length/@Split[#]>2&]],{n,0,30}]

A330028 Number of compositions of n with cuts-resistance <= 2.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 23, 45, 86, 159, 303, 568, 1069, 2005, 3769, 7066, 13251, 24821, 46482, 86988, 162758

Offset: 0

Gus Wiseman, Nov 27 2019

A composition of n is a finite sequence of positive integers summing to n.

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)    (3)    (4)      (5)
           (1,1)  (1,2)  (1,3)    (1,4)
                  (2,1)  (2,2)    (2,3)
                         (3,1)    (3,2)
                         (1,1,2)  (4,1)
                         (1,2,1)  (1,1,3)
                         (2,1,1)  (1,2,2)
                                  (1,3,1)
                                  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,2,1)
                                  (1,2,1,1)

Sum of first three columns of A329861.
Compositions with cuts-resistance 1 are A003242.
Compositions with cuts-resistance 2 are A329863.
Compositions with runs-resistance 2 are A329745.
Numbers whose binary expansion has cuts-resistance 2 are A329862.
Binary words with cuts-resistance 2 are A027383.
Cuts-resistance of binary expansion is A319416.
Binary words counted by cuts-resistance are A319421 or A329860.
Cf. A000975, A003242, A032020, A114901, A240085, A261983, A319420, A329738, A329744, A329864.

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],degdep[#]<=2&]],{n,0,10}]

A373955 Numbers k such that the k-th integer composition in standard order contains two adjacent ones and no other runs.

Original entry on oeis.org

3, 11, 14, 19, 27, 28, 29, 35, 46, 51, 56, 57, 67, 75, 78, 83, 91, 92, 93, 99, 110, 112, 113, 114, 116, 118, 131, 139, 142, 155, 156, 157, 163, 179, 184, 185, 195, 203, 206, 211, 219, 220, 221, 224, 225, 226, 229, 230, 232, 233, 236, 237, 259, 267, 270, 275

Offset: 1

Gus Wiseman, Jun 29 2024

nonn

Also numbers k such that the excess compression of the k-th integer composition in standard order is 1.
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.
postn of 1 in

			The terms and corresponding compositions begin:
    3: (1,1)
   11: (2,1,1)
   14: (1,1,2)
   19: (3,1,1)
   27: (1,2,1,1)
   28: (1,1,3)
   29: (1,1,2,1)
   35: (4,1,1)
   46: (2,1,1,2)
   51: (1,3,1,1)
   56: (1,1,4)
   57: (1,1,3,1)
   67: (5,1,1)
   75: (3,2,1,1)
   78: (3,1,1,2)
   83: (2,3,1,1)
   91: (2,1,2,1,1)
   92: (2,1,1,3)
   93: (2,1,1,2,1)
   99: (1,4,1,1)

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

These compositions are counted by A373950.
Positions of ones in A373954.
A003242 counts compressed compositions (or anti-runs).
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373948 encodes compression using compositions in standard order.
A373949 counts compositions by compression-sum.
A373953 gives compression-sum of standard compositions.
Cf. A106356, A124762, A238130, A238279, A238343, A285981, A333382, A333489, A373951, A373952.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[100],Total[stc[#]] == Total[First/@Split[stc[#]]]+1&]

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A353508 Number of integer compositions of n with no ones or runs of length 1.

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A329869 Number of compositions of n with runs-resistance equal to cuts-resistance minus 1.

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A353501 Number of integer partitions of n with all parts and all multiplicities > 2.

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A330028 Number of compositions of n with cuts-resistance <= 2.

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A373955 Numbers k such that the k-th integer composition in standard order contains two adjacent ones and no other runs.

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