A349801 -id:A349801 - cp's OEIS frontend

A349060 Number of integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.

1, 1, 2, 3, 5, 7, 10, 13, 18, 22, 29, 35, 45, 53, 68, 77, 98, 112, 140, 157, 195, 218, 270, 298, 367, 404, 495, 542, 658, 721, 873, 949, 1145, 1245, 1494, 1615, 1934, 2091, 2492, 2688, 3188, 3436, 4068, 4369, 5155, 5537, 6511, 6976, 8186, 8763, 10251, 10962

Offset: 0

Gus Wiseman, Dec 06 2021

nonn

Also the number of weakly alternating integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict increases are allowed.

			The a(1) = 1 through a(7) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (411)     (331)
                            (11111)  (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

John Tyler Rascoe, Table of n, a(n) for n = 0..300

Alternating: A025047, ranked by A345167, also A025048 and A025049.
The strong case is A065033, ranked by A167171.
A directed version is A096441.
Non-alternating: A345192, ranked by A345168.
Weakly alternating: A349052, also A129852 and A129853.
Non-weakly alternating: A349053, ranked by A349057.
A version for ordered factorizations is A349059, strong A348610.
The complement is counted by A349061, strong A349801.
These partitions are ranked by the complement of A349794.
The non-strict case is A349795.
A000041 counts integer partitions, ordered A011782.
A001250 counts alternating permutations, complement A348615.
A344604 counts alternating compositions with twins.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
Cf. A000070, A002865, A003242, A027383, A114901, A117298, A117989, A274230, A345163, A349056, A349796.

Table[Length[Select[IntegerPartitions[n], SameQ@@#||And@@EvenQ/@Take[Length/@Split[#],{2,-2}]&]],{n,0,30}]

A_x(N)={my(x='x+O('x^N), g= 1 + sum(i=1, N, (x^i/(1-x^i)) * (1 + sum(j=i+1, N-i, (x^j/((1-x^j))) / prod(k=1, j-i-1, 1-x^(2*(i+k)))))));
Vec(g)}
A_x(52) \\ John Tyler Rascoe, Mar 20 2024

G.f.: 1 + Sum_{i>0} (x^i/(1-x^i)) * (1 + Sum_{j>i} (x^j/(1-x^j)) / Product_{k=1..j-i-1} (1-x^(2*(i+k)))). - John Tyler Rascoe, Mar 20 2024

A349061 Number of integer partitions of n with at least one part of odd multiplicity that is not the first or last.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 4, 8, 13, 21, 32, 48, 67, 99, 133, 185, 245, 333, 432, 574, 732, 957, 1208, 1554, 1941, 2468, 3060, 3844, 4731, 5893, 7204, 8898, 10816, 13268, 16043, 19546, 23523, 28497, 34150, 41147, 49106, 58892, 70020, 83597, 99047, 117778, 139087

Offset: 0

Gus Wiseman, Dec 06 2021

nonn

Also the number of non-weakly alternating integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict increases are allowed.

			The a(6) = 1 through a(10) = 13 partitions:
  (321)  (421)   (431)    (432)     (532)
         (3211)  (521)    (531)     (541)
                 (4211)   (621)     (631)
                 (32111)  (3321)    (721)
                          (4311)    (4321)
                          (5211)    (5311)
                          (42111)   (6211)
                          (321111)  (32221)
                                    (33211)
                                    (43111)
                                    (52111)
                                    (421111)
                                    (3211111)

The strong version for compositions is A345192, ranked by A345168.
The version for compositions is A349053, ranked by A349057.
The complement is counted by A349060.
These partitions are ranked by A349794.
The non-strict case is A349796, complement A349795.
The strong case is A349801.
A000041 counts integer partitions.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions.
A025047 counts alternating compositions, ranked by A345167.
A025048 and A025049 count directed alternating compositions.
A096441 counts weakly alternating 0-appended partitions.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349052 counts weakly alternating compositions.
A349056 counts weakly alternating permutations of prime indices.
A349798 counts weakly but not strongly alternating perms of prime indices.
Cf. A002865, A027383, A117298, A117989, A129852, A129853, A274230, A333213, A344615, A345165, A349059.

Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&!And@@EvenQ/@Take[Length/@Split[#],{2,-2}]&]],{n,0,30}]

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

Original entry on oeis.org

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

A349795 Number of non-strict integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 7, 9, 14, 17, 24, 29, 39, 46, 61, 69, 90, 103, 131, 147, 185, 207, 259, 286, 355, 391, 482, 528, 644, 706, 858, 933, 1129, 1228, 1477, 1597, 1916, 2072, 2473, 2668, 3168, 3415, 4047, 4347, 5133, 5514, 6488, 6952, 8162, 8738, 10226, 10936

Offset: 0

Gus Wiseman, Dec 06 2021

nonn

Also the number of weakly alternating non-strict integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict increases are allowed. Equivalently, these are partitions that are weakly alternating but not strongly alternating.

			The a(2) = 1 through a(8) = 14 partitions:
  (11)  (111)  (22)    (221)    (33)      (322)      (44)
               (211)   (311)    (222)     (331)      (332)
               (1111)  (2111)   (411)     (511)      (422)
                       (11111)  (2211)    (2221)     (611)
                                (3111)    (4111)     (2222)
                                (21111)   (22111)    (3221)
                                (111111)  (31111)    (3311)
                                          (211111)   (5111)
                                          (1111111)  (22211)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

This is the restriction of A349060 to non-strict partitions.
The complement in non-strict partitions is A349796.
Permutations of prime factors of this type are counted by A349798.
The ordered version (compositions) is A349800, ranked by A349799.
These partitions are ranked by A350137.
A000041 counts integer partitions, non-strict A047967.
A001250 counts alternating permutations, complement A348615.
A025047 counts alternating compositions, also A025048 and A025049.
A096441 counts weakly alternating 0-appended partitions.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349053 counts non-weakly alternating compositions, complement A349052.
A349061 counts non-weakly alternating partitions, ranked by A349794.
A349801 counts non-alternating partitions.
Cf. A000070, A003242, A027383, A065033, A117298, A129852, A274230, A344614, A344615, A345165, A345167, A347548, A349056, A349057.

Table[Length[Select[IntegerPartitions[n],!UnsameQ@@#&&(SameQ@@#||And@@EvenQ/@Take[Length/@Split[#],{2,-2}])&]],{n,0,30}]

a(n > 0) = A349060(n) - A065033(n) = A349060(n) - floor(n/2).

a(n) = A047967(n) - A349796(n).

A349796 Number of non-strict integer partitions of n with at least one part of odd multiplicity that is not the first or last.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 8, 15, 23, 37, 52, 80, 109, 156, 208, 289, 378, 509, 654, 865, 1098, 1425, 1789, 2290, 2852, 3603, 4450, 5569, 6830, 8467, 10321, 12701, 15393, 18805, 22678, 27535, 33057, 39908, 47701, 57304, 68226, 81572, 96766, 115212, 136201

Offset: 0

Gus Wiseman, Dec 25 2021

nonn

Also the number of non-weakly alternating non-strict integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence involves the somewhat degenerate case where no strict increases are allowed.

			The a(7) = 1 through a(11) = 15 partitions:
  (3211)  (4211)   (3321)    (5311)     (4322)
          (32111)  (4311)    (6211)     (4421)
                   (5211)    (32221)    (5411)
                   (42111)   (33211)    (6311)
                   (321111)  (43111)    (7211)
                             (52111)    (42221)
                             (421111)   (43211)
                             (3211111)  (53111)
                                        (62111)
                                        (322211)
                                        (332111)
                                        (431111)
                                        (521111)
                                        (4211111)
                                        (32111111)

Counting all non-strict partitions gives A047967.
Signatures of this type are counted by A274230, complement A027383.
The strict instead of non-strict version is A347548, ranked by A350352.
The version for compositions allowing strict is A349053, ranked by A349057.
Allowing strict partitions gives A349061, complement A349060.
The complement in non-strict partitions is A349795.
These partitions are ranked by A350140 = A349794 \ A005117.
A000041 = integer partitions, strict A000009.
A001250 = alternating permutations, complement A348615.
A003242 = Carlitz (anti-run) compositions.
A025047 = alternating compositions, ranked by A345167.
A025048/A025049 = directed alternating compositions.
A096441 = weakly alternating 0-appended partitions.
A345170 = partitions w/ an alternating permutation, ranked by A345172.
A349052 = weakly alternating compositions.
A349056 = weakly alternating permutations of prime indices.
A349798 = weakly but not strongly alternating permutations of prime indices.
Cf. A000111, A002865, A117298, A117989, A129852, A129853, A345165, A345192, A349054, A349059, A349801.

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[IntegerPartitions[n],!whkQ[#]&&!whkQ[-#]&&!UnsameQ@@#&]],{n,0,30}]

a(n) = A349061(n) - A347548(n).

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.

A349794 Numbers whose prime signature has an odd term other than the first or last.

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 102, 105, 110, 114, 120, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 182, 186, 190, 195, 204, 210, 220, 222, 228, 230, 231, 238, 240, 246, 255, 258, 260, 264, 266, 270, 273, 276, 280, 282, 285, 286, 290, 294, 300, 308

Offset: 1

Gus Wiseman, Dec 06 2021

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Also numbers whose multiset of prime factors is not weakly alternating, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict decreases are allowed.

			The terms and their prime indices begin:
   30: {1,2,3}
   42: {1,2,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
  102: {1,2,7}
  105: {2,3,4}
  110: {1,3,5}
  114: {1,2,8}
  120: {1,1,1,2,3}
  130: {1,3,6}
  132: {1,1,2,5}
  138: {1,2,9}

The complement for compositions is A025047, ranked by A345167.
Signatures of this type are counted by A274230, complement A027383.
The strong case is A289553, complement A167171.
The strong case for compositions is A345192, ranked by A345168.
The version for compositions is A349053, ranked by A349057.
These partitions are counted by A349061, complement A349060, strong A349801.
The non-strict case is counted by A349795.
A001250 counts alternating permutations, complement A348615.
A096441 counts weakly alternating partitions if 0 is appended.
A345164 counts alternating permutations of prime indices, weak A349056.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349052 counts weakly alternating compositions.
A349059 counts weakly alternating ordered factorizations, strong A348610.
Cf. A003242, A112798, A114901, A117298, A128761, A129852, A129853, A344614, A344615, A345165, A349796, A349798.

Select[Range[100],PrimeNu[#]>1&&!And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}]&]

A350251 Number of non-alternating permutations of the multiset of prime factors of n.

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, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 2, 0, 2, 2, 0, 0, 5, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 8, 0, 0, 2, 1, 0, 2, 0, 2, 0, 2, 0, 9, 0, 0, 2, 2, 0, 2, 0, 5, 1, 0, 0, 8, 0, 0, 0

Offset: 1

Gus Wiseman, Jan 08 2022

nonn

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).

			The a(n) permutations for selected n:
n = 4    12    24     48      60     72      90     96       120
   ----------------------------------------------------------------
    22   223   2223   22223   2235   22233   2335   222223   22235
         322   2232   22232   2253   22323   2353   222232   22253
               2322   22322   2352   22332   2533   222322   22325
               3222   23222   2532   23223   3235   223222   22352
                      32222   3225   23322   3325   232222   22523
                              3522   32223   3352   322222   22532
                              5223   32232   3532            23225
                              5322   32322   5233            23522
                                     33222   5323            25223
                                             5332            25322
                                                             32225
                                                             32252
                                                             32522
                                                             35222
                                                             52223
                                                             52232
                                                             52322
                                                             53222

The non-anti-run case is A336107, complement A335452.
The complement is counted by A345164, with twins A344606.
Positions of nonzero terms are A345171, counted by A345165.
Positions of zeros are A345172, counted by A345170.
Compositions of this type are counted by A345192, ranked by A345168.
Ordered factorizations of this type counted by A348613, complement A348610.
Compositions weakly of this type are counted by A349053, ranked by A349057.
The weak version is A349797, complement A349056.
The case that is also weakly alternating is A349798, compositions A349800.
Patterns of this type are counted by A350252, complement A345194.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions.
A008480 counts permutations of prime factors (ordered prime factorizations).
A025047/A025048/A025049 count alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798 (row lengths A001222).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344616 gives the alternating sum of prime indices, reverse A316524.
A349052/A129852/A129853 count weakly alternating compositions.
Cf. A071321, A289553, A316523, A344652, A344653, A344742, A345173, A348379, A349061, A349801, A350138, A350139.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]] ==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Permutations[Flatten[ ConstantArray@@@FactorInteger[n]]],!wigQ[#]&]],{n,100}]

a(n) = A008480(n) - A345164(n).

A350140 Nonsquarefree numbers whose prime signature has at least one odd part other the first or last.

Original entry on oeis.org

60, 84, 120, 132, 140, 150, 156, 168, 204, 220, 228, 240, 260, 264, 270, 276, 280, 294, 300, 308, 312, 315, 336, 340, 348, 364, 372, 378, 380, 408, 420, 440, 444, 456, 460, 476, 480, 490, 492, 495, 516, 520, 528, 532, 540, 552, 560, 564, 572, 580, 585, 588

Offset: 1

Gus Wiseman, Dec 25 2021

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Also Heinz numbers of non-weakly alternating non-strict integer partitions, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. These partitions are counted by A349796. This sequence involves the somewhat degenerate case where no strict increases are allowed.

			The terms together with their Heinz partitions begin (A-E = 10-14):
     60: (3211)      276: (9211)      420: (43211)
     84: (4211)      280: (43111)     440: (53111)
    120: (32111)     294: (4421)      444: (C211)
    132: (5211)      300: (33211)     456: (82111)
    140: (4311)      308: (5411)      460: (9311)
    150: (3321)      312: (62111)     476: (7411)
    156: (6211)      315: (4322)      480: (3211111)
    168: (42111)     336: (421111)    490: (4431)
    204: (7211)      340: (7311)      492: (D211)
    220: (5311)      348: (A211)      495: (5322)
    228: (8211)      364: (6411)      516: (E211)
    240: (321111)    372: (B211)      520: (63111)
    260: (6311)      378: (42221)     528: (521111)
    264: (52111)     380: (8311)      532: (8411)
    270: (32221)     408: (72111)     540: (322211)

Including all nonsquarefree numbers gives A013929, complement A005117.
Subsets include A088860 and A110286.
Signatures of this type are counted by A274230, complement A027383.
The strict instead of non-strict version is A336568, counted by A347548.
A version for compositions allowing strict is A349057, counted by A349053.
Allowing strict partitions gives A349794, counted by A349061.
These partitions are counted by A349796.
The complement in nonsquarefree partitions is A350137, counted by A349795.
A000041 = integer partitions, strict A000009.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A096441 = weakly alternating 0-appended partitions.
A124010 = prime signature, sorted A118914.
A345164 = alternating permutations of prime indices, complement A350251.
A345170 = partitions w/ an alternating permutation, ranked by A345172.
A349052/A129852/A129853 = weakly alternating compositions.
A349056 = weakly alternating permutations of prime indices.
A349058 = weakly alternating patterns, complement A350138.
A349060 = weakly alternating partitions, strong A349801.
A349798 = weakly but not strongly alternating perms of prime indices.
Cf. A000111, A047967, A333213, A335448, A344615, A344653, A345173, A349054, A349059, A349797, A349799.

Select[Range[300],!SquareFreeQ[#]&&PrimeNu[#]>1&& !And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}]&]

Complement of A005117 in A349794.

cp's OEIS Frontend

A349060 Number of integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.

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A349061 Number of integer partitions of n with at least one part of odd multiplicity that is not the first or last.

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

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A349795 Number of non-strict integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.

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A349796 Number of non-strict integer partitions of n with at least one part of odd multiplicity that is not the first or last.

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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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A349794 Numbers whose prime signature has an odd term other than the first or last.

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