A345195 -id:A345195 - cp's OEIS frontend

A349800 Number of integer compositions of n that are weakly alternating and have at least two adjacent equal parts.

0, 0, 1, 1, 4, 9, 16, 33, 62, 113, 205, 373, 664, 1190, 2113, 3744, 6618, 11683, 20564, 36164, 63489, 111343, 195042, 341357, 596892, 1042976, 1821179, 3178145, 5543173, 9663545, 16839321, 29332231, 51075576, 88908912, 154722756, 269186074, 468221264

Offset: 0

Gus Wiseman, Dec 16 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

This sequence counts compositions that are weakly but not strongly alternating; also weakly alternating non-anti-run compositions.

			The a(2) = 1 through a(6) = 16 compositions:
  (1,1)  (1,1,1)  (2,2)      (1,1,3)      (3,3)
                  (1,1,2)    (1,2,2)      (1,1,4)
                  (2,1,1)    (2,2,1)      (2,2,2)
                  (1,1,1,1)  (3,1,1)      (4,1,1)
                             (1,1,1,2)    (1,1,1,3)
                             (1,1,2,1)    (1,1,2,2)
                             (1,2,1,1)    (1,1,3,1)
                             (2,1,1,1)    (1,3,1,1)
                             (1,1,1,1,1)  (2,2,1,1)
                                          (3,1,1,1)
                                          (1,1,1,1,2)
                                          (1,1,1,2,1)
                                          (1,1,2,1,1)
                                          (1,2,1,1,1)
                                          (2,1,1,1,1)
                                          (1,1,1,1,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Wikipedia, Alternating permutation

This is the weakly alternating case of A345192, ranked by A345168.
The case of partitions is A349795, ranked by A350137.
The version counting permutations of prime indices is A349798.
These compositions are ranked by A349799.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions, ranked by A333489.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A261983 = non-anti-run compositions, ranked by A348612.
A345165 = partitions without an alternating permutation, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
A345173 = non-alternating anti-run partitions, ranked by A345166.
A345195 = non-alternating anti-run compositions, ranked by A345169.
A348377 = non-alternating non-twin compositions.
A349801 = non-alternating partitions, ranked by A289553.
Weakly alternating:
- A349052 = compositions, directed A129852/A129853, complement A349053.
- A349056 = permutations of prime indices, complement A349797.
- A349057 = complement of standard composition numbers (too dense).
- A349058 = patterns, complement A350138.
- A349059 = ordered factorizations, complement A350139.
- A349060 = partitions, complement A349061.
Cf. A008965, A011782, A027383, A096441, A274230, A333213, A344614, A344615, A348382, A348613, A349796, A350140.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y] &&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],(whkQ[#]||whkQ[-#])&&!wigQ[#]&]],{n,0,10}]

a(n) = A349052(n) - A025047(n). - Andrew Howroyd, Jan 31 2024

a(21) onwards from Andrew Howroyd, Jan 31 2024

A348609 Numbers with a separable factorization without an alternating permutation.

Original entry on oeis.org

216, 270, 324, 378, 432, 486, 540, 594, 640, 648, 702, 756, 768, 810, 864, 896, 918, 960, 972, 1024, 1026, 1080, 1134, 1152, 1188, 1242, 1280, 1296, 1344, 1350, 1404, 1408, 1458, 1500, 1512, 1536, 1566, 1620, 1664, 1674, 1728, 1750, 1782, 1792, 1836, 1890

Offset: 1

Gus Wiseman, Oct 30 2021

nonn

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts. Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of the remaining multiplicities plus one.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of sets.
Note that 216 has separable prime factorization (2*2*2*3*3*3) with an alternating permutation, but the separable factorization (2*3*3*3*4) is has no alternating permutation. See also A345173.

			The terms and their prime factorizations begin:
  216 = 2*2*2*3*3*3
  270 = 2*3*3*3*5
  324 = 2*2*3*3*3*3
  378 = 2*3*3*3*7
  432 = 2*2*2*2*3*3*3
  486 = 2*3*3*3*3*3
  540 = 2*2*3*3*3*5
  594 = 2*3*3*3*11
  640 = 2*2*2*2*2*2*2*5
  648 = 2*2*2*3*3*3*3
  702 = 2*3*3*3*13
  756 = 2*2*3*3*3*7
  768 = 2*2*2*2*2*2*2*2*3
  810 = 2*3*3*3*3*5
  864 = 2*2*2*2*2*3*3*3

Partitions of this type are counted by A345166, ranked by A345173 (a superset).
Compositions of this type are counted by A345195, ranked by A345169.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A025047 counts alternating compositions, complement A345192, ranked by A345167.
A335434 counts separable factorizations, with twins A348383, complement A333487.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A345165 counts partitions w/o an alternating permutation, complement A345170.
A347438 counts factorizations with alternating product 1, additive A119620.
A348379 counts factorizations w/ an alternating permutation, complement A348380.
A348610 counts alternating ordered factorizations, complement A348613.
Cf. A001248, A003242, A038548, A325534, A336103, A344614, A345162, A347050, A347437, A347706, A348377, A348381, A348382.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sepQ[m_]:=Select[Permutations[m],!MatchQ[#,{_,x_,x_,_}]&]!={};
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[1000],Function[n,Select[facs[n],sepQ[#]&&Select[Permutations[#],wigQ]=={}&]!={}]]

A349798 Number of weakly alternating ordered prime factorizations of n with at least two adjacent equal parts.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 5, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 2, 2, 0, 0, 0, 5, 1, 0, 0, 2, 0, 0, 0

Offset: 1

Gus Wiseman, Dec 14 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence counts permutations of prime factors that are weakly but not strongly alternating. Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.

			Using prime indices instead of factors, the a(n) ordered prime factorizations for selected n are:
n = 4    12    24     48      90     120     192       240      270
   ------------------------------------------------------------------
    11   112   1112   11112   1223   11132   1111112   111132   12232
         211   1121   11121   1322   11213   1111121   111213   13222
               1211   11211   2213   11312   1111211   111312   21223
               2111   12111   2231   21113   1112111   112131   21322
                      21111   3122   21311   1121111   113121   22132
                              3221   23111   1211111   121113   22213
                                     31112   2111111   121311   22231
                                     31211             131112   22312
                                                       131211   23122
                                                       211131   23221
                                                       213111   31222
                                                       231111   32212
                                                       311121
                                                       312111

Wikipedia, Alternating permutation

This is the weakly but not strictly alternating case of A008480.
Including alternating (in fact, anti-run) permutations gives A349056.
These partitions are counted by A349795, ranked by A350137.
A complementary version is A349796, ranked by A350140.
The version for compositions is A349800, ranked by A349799.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A335452 = anti-run ordered prime factorizations.
A344652 = ordered prime factorizations w/o weakly increasing triples.
A345164 = alternating ordered prime factorizations, with twins A344606.
A345194 = alternating patterns, with twins A344605.
A349052/A129852/A129853 = weakly alternating compositions.
A349053 = non-weakly alternating compositions, ranked by A349057.
A349060 = weakly alternating partitions, complement A349061.
A349797 = non-weakly alternating ordered prime factorizations.
Cf. A005649, A096441, A335433, A335448, A344614, A344615, A345166, A345173, A345195, A348609, A349794, A349801.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Permutations[primeMS[n]],(whkQ[#]||whkQ[-#])&&MatchQ[#,{_,x_,x_,_}]&]],{n,100}]

A349799 Numbers k such that the k-th composition in standard order is weakly alternating but has at least two adjacent equal parts.

Original entry on oeis.org

3, 7, 10, 11, 14, 15, 19, 21, 23, 26, 27, 28, 29, 30, 31, 35, 36, 39, 42, 43, 47, 51, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 79, 83, 84, 85, 86, 87, 90, 91, 94, 95, 99, 100, 103, 106, 111, 112, 113, 114, 115, 118, 119, 120, 121, 122, 123, 124, 125

Offset: 1

Gus Wiseman, Dec 15 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
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.
This sequence ranks compositions that are weakly but not strongly alternating.

			The terms and corresponding compositions begin:
   3: (1,1)
   7: (1,1,1)
  10: (2,2)
  11: (2,1,1)
  14: (1,1,2)
  15: (1,1,1,1)
  19: (3,1,1)
  21: (2,2,1)
  23: (2,1,1,1)
  26: (1,2,2)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  31: (1,1,1,1,1)

Wikipedia, Alternating permutation

Partitions of this type are counted by A349795, ranked by A350137.
Permutations of prime indices of this type are counted by A349798.
These compositions are counted by A349800.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions, ranked by A333489.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A261983 = non-anti-run compositions, ranked by A348612.
A345164 = alternating permutations of prime indices, with twins A344606.
A345165 = partitions without an alternating permutation, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
A345166 = separable partitions with no alternations, ranked by A345173.
A345192 = non-alternating compositions, ranked by A345168.
A345195 = non-alternating anti-run compositions, ranked by A345169.
A349052/A129852/A129853 = weakly alternating compositions.
A349053 = non-weakly alternating compositions, ranked by A349057.
A349056 = weak alternations of prime indices, complement A349797.
A349060 = weak alternations of partitions, complement A349061.
Cf. A005649, A049774, A096441, A128761, A344615, A345163, A349054, A349058, A349796, A349801, A350140.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Select[Range[0,100],(whkQ[stc[#]]||whkQ[-stc[#]])&&MatchQ[stc[#],{_,x_,x_,_}]&]

Equals A345168 \ A349057 = A348612 \ A349057.

A348611 Number of ordered factorizations of n with no adjacent equal factors.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 14, 1, 3, 3, 6, 1, 13, 1, 7, 3, 3, 3, 17, 1, 3, 3, 14, 1, 13, 1, 6, 6, 3, 1, 29, 1, 6, 3, 6, 1, 14, 3, 14, 3, 3, 1, 36, 1, 3, 6, 14, 3, 13, 1, 6, 3, 13, 1, 45, 1, 3, 6, 6, 3, 13, 1, 29, 4, 3

Offset: 1

Gus Wiseman, Nov 07 2021

nonn

First differs from A348610 at a(24) = 14, A348610(24) = 12.
An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
In analogy with Carlitz compositions, these may be called Carlitz ordered factorizations.

			The a(n) ordered factorizations without adjacent equal factors for n = 1, 6, 12, 16, 24, 30, 32, 36 are:
  ()   6     12      16      24      30      32      36
       2*3   2*6     2*8     3*8     5*6     4*8     4*9
       3*2   3*4     8*2     4*6     6*5     8*4     9*4
             4*3     2*4*2   6*4     10*3    16*2    12*3
             6*2             8*3     15*2    2*16    18*2
             2*3*2           12*2    2*15    2*8*2   2*18
                             2*12    3*10    4*2*4   3*12
                             2*3*4   2*3*5           2*3*6
                             2*4*3   2*5*3           2*6*3
                             2*6*2   3*2*5           2*9*2
                             3*2*4   3*5*2           3*2*6
                             3*4*2   5*2*3           3*4*3
                             4*2*3   5*3*2           3*6*2
                             4*3*2                   6*2*3
                                                     6*3*2
                                                     2*3*2*3
                                                     3*2*3*2
Thus, of total A074206(12) = 8 ordered factorizations of 12, only factorizations 2*2*3 and 3*2*2 (see A348616) are not included in this count, therefore a(12) = 6. - _Antti Karttunen_, Nov 12 2021

The additive version (compositions) is A003242, complement A261983.
The additive alternating version is A025047, ranked by A345167.
Factorizations without a permutation of this type are counted by A333487.
As compositions these are ranked by A333489, complement A348612.
Factorizations with a permutation of this type are counted by A335434.
The non-alternating additive version is A345195, ranked by A345169.
The alternating case is A348610, which is dominated at positions A122181.
The complement is counted by A348616.
A001055 counts factorizations, strict A045778, ordered A074206.
A325534 counts separable partitions, ranked by A335433.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A348613 counts non-alternating ordered factorizations.
Cf. A001250, A138364, A336103, A347050, A347438, A347463, A347466, A347706, A348379, A348383.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
antirunQ[y_]:=Length[y]==Length[Split[y]]
Table[Length[Select[ordfacs[n],antirunQ]],{n,100}]

A348611(n, e=0) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d!=e), s += A348611(n/d, d))); (s)); \\ Antti Karttunen, Nov 12 2021

a(n) = A074206(n) - A348616(n).

A348382 Number of compositions of n that are not a twin (x,x) but have adjacent equal parts.

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 17, 41, 88, 185, 387, 810, 1669, 3435, 7039, 14360, 29225, 59347, 120228, 243166, 491085, 990446, 1995409, 4016259, 8076959, 16231746, 32599773, 65437945, 131293191, 263316897, 527912139, 1058061751, 2120039884, 4246934012, 8505864639

Offset: 0

Gus Wiseman, Nov 05 2021

nonn

A composition with no adjacent equal parts is also called a Carlitz composition, so these are non-twin, non-Carlitz compositions.

			The a(3) = 1 through a(6) = 17 compositions:
  (111)  (112)   (113)    (114)
         (211)   (122)    (222)
         (1111)  (221)    (411)
                 (311)    (1113)
                 (1112)   (1122)
                 (1121)   (1131)
                 (1211)   (1221)
                 (2111)   (1311)
                 (11111)  (2112)
                          (2211)
                          (3111)
                          (11112)
                          (11121)
                          (11211)
                          (12111)
                          (21111)
                          (111111)

A. Knopfmacher and H. Prodinger, On Carlitz Compositions, Europ. J. Combinatorics (1998) 19, 579-589.
Wikipedia, Composition (combinatorics)

Allowing twins gives A261983, complement A003242.
The non-alternating case is A348377, difference A345195.
These compositions are ranked by A348612 \ A007582.
A001250 counts alternating permutations, complement A348615.
A007582 ranks twin compositions.
A011782 counts compositions, strict A032020.
A025047 counts alternating or wiggly compositions, complement A345192.
A051049 counts non-twin compositions, complement A000035(n+1).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A000070, A005649, A059841, A106356, A238279, A333755, A344604, A344614, A344740, A348381.

nn=15;CoefficientList[Series[1+x/(1-2x)-x^2/(1-x^2)-1/(1-Sum[x^k/(1+x^k),{k,1,nn}]),{x,0,nn}],x]

For n > 0, a(n) = A261983(n) - A059841(n).

O.g.f.: 1 + x/(1-2x) - x^2/(1-x^2) - 1/(1 - Sum_{k>0} x^k/(1+x^k)).

A129838 Number of up/down (or down/up) compositions of n into distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 18, 21, 30, 38, 52, 78, 97, 128, 170, 222, 285, 421, 510, 683, 872, 1148, 1440, 1893, 2576, 3209, 4151, 5313, 6784, 8615, 10969, 13755, 18573, 22713, 29173, 36536, 46705, 57899, 73696, 91076, 114777, 148531, 182813, 228938, 287042

Offset: 0

Vladeta Jovovic, May 21 2007

Original name was: Number of alternating compositions of n into distinct parts.

A composition is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. - Gus Wiseman, Jan 15 2022

			From _Gus Wiseman_, Jan 15 2022: (Start)
The a(1) = 1 through a(8) = 8 up/down strict compositions (non-strict A025048):
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)
                          (2,3)  (2,4)    (2,5)    (2,6)
                                 (1,3,2)  (3,4)    (3,5)
                                 (2,3,1)  (1,4,2)  (1,4,3)
                                          (2,4,1)  (1,5,2)
                                                   (2,5,1)
                                                   (3,4,1)
The a(1) = 1 through a(8) = 8 down/up strict compositions (non-strict A025049):
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)    (5,3)
                          (4,1)  (5,1)    (5,2)    (6,2)
                                 (2,1,3)  (6,1)    (7,1)
                                 (3,1,2)  (2,1,4)  (2,1,5)
                                          (4,1,2)  (3,1,4)
                                                   (4,1,3)
                                                   (5,1,2)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

The case of permutations is A000111.
This is the up/down case of A032020.
This is the strict case of A129852/A129853, strong A025048/A025049.
The undirected version is A349054.
A001250 = alternating permutations, complement A348615.
A003242 = Carlitz compositions, complement A261983.
A011782 = compositions, unordered A000041.
A025047 = alternating compositions, complement A345192.
A349052 = weakly alternating compositions, complement A349053.
Cf. A003056, A008289, A008965, A015723, A072706, A128761, A218074, A345165, A345170, A345195, A349800.

g:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(g(o-1+j, u-j), j=1..u))
    end:
b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), b(n-k, k)+b(n-k, k-1)))
    end:
a:= n-> add(b(n, k)*g(k, 0), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 22 2021

whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@ Select[IntegerPartitions[n],UnsameQ@@#&],whkQ]],{n,0,15}] (* Gus Wiseman, Jan 15 2022 *)

G.f.: Sum_{k>=0} A000111(k)*x^(k*(k+1)/2)/Product_{i=1..k} (1-x^i). - Vladeta Jovovic, May 24 2007
a(n) = Sum_{k=0..A003056(n)} A000111(k) * A008289(n,k). - Alois P. Heinz, Dec 22 2021
a(n) = (A349054(n) + 1)/2. - Gus Wiseman, Jan 15 2022

a(0)=1 prepended by Alois P. Heinz, Dec 22 2021

Name changed from "alternating" to "up/down" by Gus Wiseman, Jan 15 2022

cp's OEIS Frontend

A349800 Number of integer compositions of n that are weakly alternating and have at least two adjacent equal parts.

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A348609 Numbers with a separable factorization without an alternating permutation.

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A349798 Number of weakly alternating ordered prime factorizations of n with at least two adjacent equal parts.

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A349799 Numbers k such that the k-th composition in standard order is weakly alternating but has at least two adjacent equal parts.

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A348611 Number of ordered factorizations of n with no adjacent equal factors.

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A348382 Number of compositions of n that are not a twin (x,x) but have adjacent equal parts.

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A129838 Number of up/down (or down/up) compositions of n into distinct parts.

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Maple

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