A344616 -id:A344616 - cp's OEIS frontend

A344653 Every permutation of the prime factors of n has a consecutive monotone triple, i.e., a triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 80, 81, 88, 96, 104, 112, 125, 128, 135, 136, 144, 152, 160, 162, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 270, 272, 288, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351, 352, 368, 375, 376, 378, 384

Offset: 1

Gus Wiseman, Jun 12 2021

nonn

Differs from A335448 in lacking squares and having 270 etc.
First differs from A345193 in having 270.
Such a permutation is characterized by being neither a twin (x,x) nor wiggly (A025047, A345192). A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no wiggly permutations, even though it has anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
   8: {1,1,1}
  16: {1,1,1,1}
  24: {1,1,1,2}
  27: {2,2,2}
  32: {1,1,1,1,1}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  80: {1,1,1,1,3}
  81: {2,2,2,2}
  88: {1,1,1,5}
  96: {1,1,1,1,1,2}
For example, 36 has prime indices (1,1,2,2), which has the two wiggly permutations (1,2,1,2) and (2,1,2,1), so 36 is not in the sequence.

A superset of A335448, counted by A325535.
Positions of 0's in A344606.
These partitions are counted by A344654.
The complement is A344742, counted by A344740.
The separable case is A345173, counted by A345166.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001250 counts wiggly permutations.
A003242 counts anti-run compositions.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A325534 counts separable partitions, ranked by A335433.
A344604 counts wiggly compositions with twins.
A345164 counts wiggly permutations of prime indices.
A345165 counts partitions without a wiggly permutation, ranked by A345171.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions.
Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344614, A344615, A344616, A344652, A345163, A345168, A345193.

Select[Range[100],Select[Permutations[Flatten[ConstantArray@@@FactorInteger[#]]],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]=={}&]

Complement of A001248 in A345171.

A344604 Number of alternating compositions of n, including twins (x,x).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 13, 19, 30, 48, 76, 118, 187, 293, 461, 725, 1140, 1789, 2815, 4422, 6950, 10924, 17169, 26979, 42405, 66644, 104738, 164610, 258708, 406588, 639010, 1004287, 1578364, 2480606, 3898600, 6127152, 9629624, 15134213, 23785389, 37381849, 58750469

Offset: 0

Gus Wiseman, May 27 2021

nonn

We define a composition to be alternating including twins (x,x) if there are no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z. Except in the case of twins (x,x), all such compositions are anti-runs (A003242). These compositions avoid the weak consecutive patterns (1,2,3) and (3,2,1), the strict version being A344614.

The version without twins (x,x) is A025047 (alternating compositions).

			The a(1) = 1 through a(7) = 19 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)
       (11)  (12)  (13)   (14)   (15)    (16)
             (21)  (22)   (23)   (24)    (25)
                   (31)   (32)   (33)    (34)
                   (121)  (41)   (42)    (43)
                          (131)  (51)    (52)
                          (212)  (132)   (61)
                                 (141)   (142)
                                 (213)   (151)
                                 (231)   (214)
                                 (312)   (232)
                                 (1212)  (241)
                                 (2121)  (313)
                                         (412)
                                         (1213)
                                         (1312)
                                         (2131)
                                         (3121)
                                         (12121)

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

A001250 counts alternating permutations.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A325534 counts separable partitions.
A325535 counts inseparable partitions.
A344605 counts alternating patterns including twins.
A344606 counts alternating permutations of prime factors including twins.
Counting compositions by patterns:
- A011782 no conditions.
- A003242 avoiding (1,1) adjacent.
- A102726 avoiding (1,2,3).
- A106351 avoiding (1,1) adjacent by sum and length.
- A128695 avoiding (1,1,1) adjacent.
- A128761 avoiding (1,2,3) adjacent.
- A232432 avoiding (1,1,1).
- A335456 all patterns.
- A335457 all patterns adjacent.
- A335514 matching (1,2,3).
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A000041, A006330, A008965, A238279, A239830, A333213, A238279/A333755, A344612, A344616, A344617, A344618.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]],{n,0,15}]

a(n > 0) = A025047(n) + 1 if n is even, otherwise A025047(n). - Gus Wiseman, Nov 03 2021

a(21)-a(40) from Alois P. Heinz, Nov 04 2021

A344610 Triangle read by rows where T(n,k) is the number of integer partitions of 2n with reverse-alternating sum 2k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 3, 1, 1, 7, 9, 6, 3, 1, 1, 11, 14, 12, 6, 3, 1, 1, 15, 23, 20, 12, 6, 3, 1, 1, 22, 34, 35, 21, 12, 6, 3, 1, 1, 30, 52, 56, 38, 21, 12, 6, 3, 1, 1, 42, 75, 91, 62, 38, 21, 12, 6, 3, 1, 1, 56, 109, 140, 103, 63, 38, 21, 12, 6, 3, 1, 1

Offset: 0

Gus Wiseman, May 31 2021

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts.

Also the number of reversed integer partitions of 2n with alternating sum 2k.

			Triangle begins:
   1
   1   1
   2   1   1
   3   3   1   1
   5   5   3   1   1
   7   9   6   3   1   1
  11  14  12   6   3   1   1
  15  23  20  12   6   3   1   1
  22  34  35  21  12   6   3   1   1
  30  52  56  38  21  12   6   3   1   1
  42  75  91  62  38  21  12   6   3   1   1
  56 109 140 103  63  38  21  12   6   3   1   1
  77 153 215 163 106  63  38  21  12   6   3   1   1
Row n = 5 counts the following partitions:
  (55)          (442)        (433)      (622)    (811)  (10)
  (3322)        (541)        (532)      (721)
  (4411)        (22222)      (631)      (61111)
  (222211)      (32221)      (42211)
  (331111)      (33211)      (52111)
  (22111111)    (43111)      (4111111)
  (1111111111)  (2221111)
                (3211111)
                (211111111)

The columns with initial 0's removed appear to converge to A006330.
The odd version is A239829.
The non-reversed version is A239830.
Row sums are A344611, odd bisection of A344607.
Including odd n and negative k gives A344612 (strict: A344739).
The strict case is A344649 (row sums: A344650).
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum.
A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344604 counts wiggly compositions with twins.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A000070, A000097, A001250, A003242, A027187, A028260, A124754, A152146, A344608, A344651, A344654.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],k==sats[#]&]],{n,0,15,2},{k,0,n,2}]

A344619 The a(n)-th composition in standard order (A066099) has alternating sum 0.

Original entry on oeis.org

0, 3, 10, 13, 15, 36, 41, 43, 46, 50, 53, 55, 58, 61, 63, 136, 145, 147, 150, 156, 162, 165, 167, 170, 173, 175, 180, 185, 187, 190, 196, 201, 203, 206, 210, 213, 215, 218, 221, 223, 228, 233, 235, 238, 242, 245, 247, 250, 253, 255, 528, 545, 547, 550, 556, 568

Offset: 1

Gus Wiseman, Jun 03 2021

nonn

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

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.

			The sequence of terms together with the corresponding compositions begins:
    0: ()
    3: (1,1)
   10: (2,2)
   13: (1,2,1)
   15: (1,1,1,1)
   36: (3,3)
   41: (2,3,1)
   43: (2,2,1,1)
   46: (2,1,1,2)
   50: (1,3,2)
   53: (1,2,2,1)
   55: (1,2,1,1,1)
   58: (1,1,2,2)
   61: (1,1,1,2,1)
   63: (1,1,1,1,1,1)
  136: (4,4)
  145: (3,4,1)
  147: (3,3,1,1)
  150: (3,2,1,2)
  156: (3,1,1,3)

The version for Heinz numbers of partitions is A000290, counted by A000041.
These are the positions of zeros in A344618.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A116406 counts compositions with alternating sum >= 0.
A124754 gives the alternating sum of standard compositions.
A316524 is the alternating sum of the prime indices of n.
A344604 counts wiggly compositions with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A344616 gives the alternating sum of reversed prime indices.
All of the following pertain to compositions in standard order:
- The length is A000120.
- Converting to reversed ranking gives A059893.
- The rows are A066099.
- The sum is A070939.
- The runs are counted by A124767.
- The reversed version is A228351.
- Strict compositions are ranked by A233564.
- Constant compositions are ranked by A272919.
- The Heinz number is A333219.
- Anti-run compositions are ranked by A333489.
Cf. A000070, A000097, A003242, A006330, A028260, A119899, A239830, A344605, A344607, A344650, A344739.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]]
Select[Range[0,100],ats[stc[#]]==0&]

A345197 Concatenation of square matrices A(n), each read by rows, where A(n)(k,i) is the number of compositions of n of length k with alternating sum i, where 1 <= k <= n, and i ranges from -n + 2 to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 2, 3, 0, 0, 2, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 3, 4, 0, 0, 3, 4, 3, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 0

Gus Wiseman, Jul 03 2021

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The matrices for n = 1..7:
  1   0 1   0 0 1   0 0 0 1   0 0 0 0 1   0 0 0 0 0 1   0 0 0 0 0 0 1
      1 0   1 1 0   1 1 1 0   1 1 1 1 0   1 1 1 1 1 0   1 1 1 1 1 1 0
            0 1 0   0 1 2 0   0 1 2 3 0   0 1 2 3 4 0   0 1 2 3 4 5 0
                    0 1 0 0   0 2 2 0 0   0 3 4 3 0 0   0 4 6 6 4 0 0
                              0 0 1 0 0   0 0 2 3 0 0   0 0 3 6 6 0 0
                                          0 0 1 0 0 0   0 0 3 3 0 0 0
                                                        0 0 0 1 0 0 0
Matrix n = 5 counts the following compositions:
           i=-3:        i=-1:          i=1:            i=3:        i=5:
        -----------------------------------------------------------------
   k=1: |    0            0             0               0          (5)
   k=2: |   (14)         (23)          (32)            (41)         0
   k=3: |    0          (131)       (221)(122)   (311)(113)(212)    0
   k=4: |    0       (1211)(1112)  (2111)(1121)         0           0
   k=5: |    0            0          (11111)            0           0

Gus Wiseman, A raster plot of the zeros in A(16).

The number of nonzero terms in each matrix appears to be A000096.
The number of zeros in each matrix appears to be A000124.
Row sums and column sums both appear to be A007318 (Pascal's triangle).
The matrix sums are A131577.
Antidiagonal sums appear to be A163493.
The reverse-alternating version is also A345197 (this sequence).
Antidiagonals are A345907.
Traces are A345908.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Other tetrangles: A318393, A318816, A320808, A321912.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000097, A000346, A007318, A008549, A025047, A032443, A034871, A114121, A120452, A238279, A239830, A344604.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]==k&&ats[#]==i&]],{n,0,6},{k,1,n},{i,-n+2,n,2}]

A344611 Number of integer partitions of 2n with reverse-alternating sum >= 0.

Original entry on oeis.org

1, 2, 4, 8, 15, 27, 48, 81, 135, 220, 352, 553, 859, 1313, 1986, 2969, 4394, 6439, 9357, 13479, 19273, 27353, 38558, 53998, 75168, 104022, 143172, 196021, 267051, 362086, 488733, 656802, 879026, 1171747, 1555997, 2058663, 2714133, 3566122, 4670256, 6096924, 7935184

Offset: 0

Gus Wiseman, May 30 2021

nonn

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
Also the number of reversed integer partitions of 2n with alternating sum >= 0.
The reverse-alternating sum of a partition is equal to (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of partitions of 2n whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of partitions of 2n whose parts are all even or whose greatest part is odd.

			The a(0) = 1 through a(4) = 15 partitions:
  ()  (2)   (4)     (6)       (8)
      (11)  (22)    (33)      (44)
            (211)   (222)     (332)
            (1111)  (321)     (422)
                    (411)     (431)
                    (2211)    (521)
                    (21111)   (611)
                    (111111)  (2222)
                              (3311)
                              (22211)
                              (32111)
                              (41111)
                              (221111)
                              (2111111)
                              (11111111)

The non-reversed version is A058696 (partitions of 2n).
The ordered version appears to be A114121.
Odd bisection of A344607.
Row sums of A344610.
The strict case is A344650.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions with alternating sum 1.
A000097 counts partitions with alternating sum 2.
A103919 counts partitions by sum and alternating sum.
A120452 counts partitions of 2n with reverse-alternating sum 2.
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344612 counts partitions by sum and rev-alt sum (strict: A344739).
A344618 gives reverse-alternating sums of standard compositions.
A344741 counts partitions of 2n with reverse-alternating sum -2.
Cf. A001250, A027187, A028260, A116406, A119899, A124754, A152146, A239829, A344608, A344609, A344649, A344651, A344654.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]>=0&]],{n,0,30,2}]

Conjecture: a(n) <= A160786(n). The difference is 0, 0, 0, 0, 1, 2, 4, 9, 16, 28, 48, 79, ...

More terms from Bert Dobbelaere, Jun 12 2021

A345172 Numbers whose multiset of prime factors has an alternating permutation.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

First differs from A212167 in containing 72.
First differs from A335433 in lacking 270, corresponding to the partition (3,2,2,2,1).
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}          20: {1,1,3}       39: {2,6}
      2: {1}         21: {2,4}         41: {13}
      3: {2}         22: {1,5}         42: {1,2,4}
      5: {3}         23: {9}           43: {14}
      6: {1,2}       26: {1,6}         44: {1,1,5}
      7: {4}         28: {1,1,4}       45: {2,2,3}
     10: {1,3}       29: {10}          46: {1,9}
     11: {5}         30: {1,2,3}       47: {15}
     12: {1,1,2}     31: {11}          50: {1,3,3}
     13: {6}         33: {2,5}         51: {2,7}
     14: {1,4}       34: {1,7}         52: {1,1,6}
     15: {2,3}       35: {3,4}         53: {16}
     17: {7}         36: {1,1,2,2}     55: {3,5}
     18: {1,2,2}     37: {12}          57: {2,8}
     19: {8}         38: {1,8}         58: {1,10}

Including squares of primes A001248 gives A344742, counted by A344740.
This is a subset of A335433, which is counted by A325534.
Positions of nonzero terms in A345164.
The partitions with these Heinz numbers are counted by A345170.
The complement is A345171, which is counted by A345165.
A345173 = A345171 /\ A335433 is counted by A345166.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A344606 counts alternating permutations of prime indices with twins.
A345192 counts non-alternating compositions.
Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344605, A344614, A344616, A344653, A344654, A345163, A345167, A347706, A348379.

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

Complement of A001248 (squares of primes) in A344742.

A026810 Number of partitions of n in which the greatest part is 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108, 120, 136, 150, 169, 185, 206, 225, 249, 270, 297, 321, 351, 378, 411, 441, 478, 511, 551, 588, 632, 672, 720, 764, 816, 864, 920, 972, 1033, 1089, 1154, 1215, 1285, 1350

Offset: 0

Clark Kimberling

Also number of partitions of n into exactly 4 parts.
Also the number of weighted cubic graphs on 4 nodes (=the tetrahedron) with weight n. - R. J. Mathar, Nov 03 2018
From Gus Wiseman, Jun 27 2021: (Start)
Also the number of strict integer partitions of 2n with alternating sum 4, or (by conjugation) partitions of 2n covering an initial interval of positive integers with exactly 4 odd parts. The strict partitions with alternating sum 4 are:
  (4)  (5,1)  (6,2)    (7,3)    (8,4)      (9,5)      (10,6)
              (5,2,1)  (5,3,2)  (5,4,3)    (6,5,3)    (7,6,3)
                       (6,3,1)  (6,4,2)    (7,5,2)    (8,6,2)
                                (7,4,1)    (8,5,1)    (9,6,1)
                                (6,3,2,1)  (6,4,3,1)  (6,5,4,1)
                                           (7,4,2,1)  (7,4,3,2)
                                                      (7,5,3,1)
                                                      (8,5,2,1)
                                                      (6,4,3,2,1)
(End)

			From _Gus Wiseman_, Jun 27 2021: (Start)
The a(4) = 1 through a(10) = 9 partitions of length 4:
  (1111)  (2111)  (2211)  (2221)  (2222)  (3222)  (3322)
                  (3111)  (3211)  (3221)  (3321)  (3331)
                          (4111)  (3311)  (4221)  (4222)
                                  (4211)  (4311)  (4321)
                                  (5111)  (5211)  (4411)
                                          (6111)  (5221)
                                                  (5311)
                                                  (6211)
                                                  (7111)
(End)

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
D. E. Knuth, The Art of Computer Programming, vol. 4,fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.

Washington Bomfim, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

Cf. A001400, A026811, A026812, A026813, A026814, A026815, A026816, A069905 (3 positive parts), A002621 (partial sums), A005044 (first differences).
A non-strict version is A000710 or A088218.
This is column k = 2 of A152146.
A reverse version is A343941.
Cf. A000041, A000070, A000097, A067659, A103919, A120452, A236559, A239830, A306145, A343942, A344616, A344649, A344651.

[Round((n^3+3*n^2-9*n*(n mod 2))/144): n in [0..60]]; // Vincenzo Librandi, Oct 14 2015

A049347 := proc(n)
        op(1+(n mod 3),[1,-1,0]) ;
end proc:
A056594 := proc(n)
        op(1+(n mod 4),[1,0,-1,0]) ;
end proc:
A026810 := proc(n)
        1/288*(n+1)*(2*n^2+4*n-13+9*(-1)^n) ;
        %-A049347(n)/9 ;
        %+A056594(n)/8 ;
end proc: # R. J. Mathar, Jul 03 2012

Table[Count[IntegerPartitions[n], {4, _}], {n, 0, 60}]
LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, -1}, {0, 0, 0, 0, 1, 1, 2, 3, 5, 6}, 60] (* Vincenzo Librandi, Oct 14 2015 *)
Table[Length[IntegerPartitions[n, {4}]], {n, 0, 60}] (* Eric Rowland, Mar 02 2017 *)
CoefficientList[Series[x^4/Product[1 - x^k, {k, 1, 4}], {x, 0, 60}], x] (* Robert A. Russell, May 13 2018 *)

for(n=0, 60, print(n, " ", round((n^3 + 3*n^2 -9*n*(n % 2))/144))); \\ Washington Bomfim, Jul 03 2012

x='x+O('x^60); concat([0, 0, 0, 0], Vec(x^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)))) \\ Altug Alkan, Oct 14 2015

vector(60, n, n--; (n+1)*(2*n^2+4*n-13+9*(-1)^n)/288 + real(I^n)/8 - ((n+2)%3-1)/9) \\ Altug Alkan, Oct 26 2015

print1(0,", "); for(n=1,60,j=0;forpart(v=n,j++,,[4,4]); print1(j,", ")) \\ Hugo Pfoertner, Oct 01 2018

G.f.: x^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) = x^4/((1-x)^4*(1+x)^2*(1+x+x^2)*(1+x^2)).
a(n+4) = A001400(n). - Michael Somos, Apr 07 2012
a(n) = round( (n^3 + 3*n^2 -9*n*(n mod 2))/144 ). - Washington Bomfim, Jan 06 2021 and Jul 03 2012
a(n) = (n+1)*(2*n^2+4*n-13+9*(-1)^n)/288 -A049347(n)/9 +A056594(n)/8. - R. J. Mathar, Jul 03 2012
From Gregory L. Simay, Oct 13 2015: (Start)
a(n) = (n^3 + 3*n^2 - 9*n)/144 + a(m) - (m^3 + 3*m^2 - 9*m)/144 if n = 12k + m and m is odd. For example, a(23) = a(12*1 + 11) = (23^3 + 3*23^2 - 9*23)/144 + a(11) - (11^3 + 3*11^2 - 9*11)/144 = 94.
a(n) = (n^3 + 3*n^2)/144 + a(m) - (m^3 + 3*m^2)/144 if n = 12k + m and m is even. For example, a(22) = a(12*1 + 10) = (22^3 + 3*22^2)/144 + a(10) - (10^3 + 3*10^2)/144 = 84. (End)
a(n) = A008284(n,4). - Robert A. Russell, May 13 2018
From Gregory L. Simay, Jul 28 2019: (Start)
a(2n+1) = a(2n) + a(n+1) - a(n-3) and
a(2n) = a(2n-1) + a(n+2) - a(n-2). (End)

A345164 Number of alternating permutations of the multiset of prime factors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

First differs from A335452 at a(30) = 4, A335452(30) = 6. The anti-runs (2,3,5) and (5,3,2) are not alternating.

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 permutation, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The a(n) alternating permutations of prime indices for n = 180, 210, 300, 420, 900:
  (12132)  (1324)  (13132)  (12143)  (121323)
  (21213)  (1423)  (13231)  (13142)  (132312)
  (21312)  (2143)  (21313)  (13241)  (213132)
  (23121)  (2314)  (23131)  (14132)  (213231)
  (31212)  (2413)  (31213)  (14231)  (231213)
           (3142)  (31312)  (21314)  (231312)
           (3241)           (21413)  (312132)
           (3412)           (23141)  (323121)
           (4132)           (24131)
           (4231)           (31214)
                            (31412)
                            (34121)
                            (41213)
                            (41312)

Counting all permutations gives A008480.
Dominated by A335452 (number of separations of prime factors).
Including twins (x,x) gives A344606.
Positions of zeros are A345171, counted by A345165.
Positions of nonzero terms are A345172.
A000041 counts integer partitions.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A344654 counts non-twin partitions w/o alternating permutation, rank: A344653.
A344740 counts twins and partitions w/ alternating permutation, rank: A344742.
A345166 counts separable partitions w/o alternating permutation, rank: A345173.
A345170 counts partitions with a alternating permutation.
Cf. A001222, A071321, A071322, A316523, A316524, A333489, A335126, A344605, A344614, A344616, A344652, A345163, A345168.

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,30}]

A355733 Number of multisets that can be obtained by choosing a divisor of each prime index of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 4, 3, 4, 1, 2, 3, 4, 2, 5, 2, 3, 2, 3, 4, 4, 3, 4, 4, 2, 1, 4, 2, 6, 3, 6, 4, 7, 2, 2, 5, 4, 2, 6, 3, 4, 2, 6, 3, 4, 4, 5, 4, 4, 3, 7, 4, 2, 4, 6, 2, 7, 1, 7, 4, 2, 2, 6, 6, 6, 3, 4, 6, 6, 4, 6, 7, 4, 2, 5, 2, 2, 5, 4, 4, 7

Offset: 1

Gus Wiseman, Jul 16 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(15) = 4 multisets are: {1,1}, {1,2}, {1,3}, {2,3}.
The a(18) = 3 multisets are: {1,1,1}, {1,1,2}, {1,2,2}.

Robert Price, Table of n, a(n) for n = 1..2000
Wikipedia, Axiom of choice.

Counting all choices of divisors gives A355731, firsts A355732.
Positions of first appearances are A355734.
Choosing weakly increasing divisors gives A355735, firsts A355736.
Choosing only prime divisors gives A355744.
The version choosing a divisor of each number from 1 to n is A355747.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061395 selects the maximum prime index.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A340852 lists numbers that can be factored into divisors of bigomega.
Cf. A000720, A076610, A289509, A316524, A335433, A335448, A340606, A344616, A355737, A355739, A355740, A355741, A355746.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Tuples[Divisors/@primeMS[n]]]],{n,100}]

cp's OEIS Frontend

A344653 Every permutation of the prime factors of n has a consecutive monotone triple, i.e., a triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

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A344604 Number of alternating compositions of n, including twins (x,x).

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A344610 Triangle read by rows where T(n,k) is the number of integer partitions of 2n with reverse-alternating sum 2k.

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A344619 The a(n)-th composition in standard order (A066099) has alternating sum 0.

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A345197 Concatenation of square matrices A(n), each read by rows, where A(n)(k,i) is the number of compositions of n of length k with alternating sum i, where 1 <= k <= n, and i ranges from -n + 2 to n in steps of 2.

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A344611 Number of integer partitions of 2n with reverse-alternating sum >= 0.

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A345172 Numbers whose multiset of prime factors has an alternating permutation.

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A026810 Number of partitions of n in which the greatest part is 4.

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