A344654 -id:A344654 - cp's OEIS frontend

A345162 Number of integer partitions of n with no alternating permutation covering an initial interval of positive integers.

0, 0, 1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 8, 10, 11, 15, 16, 18, 23, 27, 30, 35, 41, 47, 54, 62, 71, 82, 92, 103, 121, 137, 151, 173, 195, 220, 248, 277, 311, 350, 393, 435, 488, 546, 605, 678, 754, 835, 928, 1029, 1141, 1267, 1400, 1544, 1712, 1891, 2081, 2298, 2533, 2785, 3068

Offset: 0

Gus Wiseman, Jun 12 2021

nonn

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 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).

Sequences covering an initial interval (patterns) are counted by A000670 and ranked by A333217.

			The a(2) = 1 through a(10) = 6 partitions:
  11  111  1111  2111   21111   2221     221111    22221      32221
                 11111  111111  211111   2111111   321111     222211
                                1111111  11111111  2211111    3211111
                                                   21111111   22111111
                                                   111111111  211111111
                                                              1111111111

Andrew Howroyd, Table of n, a(n) for n = 0..500

The complement in covering partitions is counted by A345163.
Not requiring normality gives A345165, ranked by A345171.
The separable case is A345166.
A000041 counts integer partitions.
A000670 counts patterns, ranked by A333217.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, directed A025048/A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A344605 counts alternating patterns with twins.
A345164 counts alternating permutations of prime indices.
A345170 counts partitions with a alternating permutation, ranked by A345172.
Cf. A000070, A103919, A335126, A344614, A344615, A344653, A344654, A344740, A344742, A345167, A345168, A345192, A348609.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&Select[Permutations[#],wigQ[#]&]=={}&]],{n,0,15}]

P(n,m)={Vec(1/prod(k=1, m, 1-y*x^k, 1+O(x*x^n)))}
a(n) = {(n >= 2) + sum(k=2, (sqrtint(8*n+1)-1)\2, my(r=n-binomial(k+1,2), v=P(r, k)); sum(i=1, min(k,2*r\k), sum(j=k-1, (2*r-(k-1)*(i-1))\(i+1), my(p=(j+k+(i==1||i==k))\2); if(p*i<=r, polcoef(v[r-p*i+1],j-p)) )))} \\ Andrew Howroyd, Jan 31 2024

a(n) = A000009(n) - A345163(n). - Andrew Howroyd, Jan 31 2024

a(26) onwards from Andrew Howroyd, Jan 31 2024

A347448 Number of integer partitions of n with alternating product > 1.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 12, 17, 25, 35, 49, 66, 90, 120, 161, 209, 275, 355, 460, 585, 750, 946, 1199, 1498, 1881, 2335, 2909, 3583, 4430, 5428, 6666, 8118, 9912, 12013, 14586, 17592, 21252, 25525, 30695, 36711, 43956, 52382, 62469, 74173, 88132, 104303, 123499

Offset: 0

Gus Wiseman, Sep 16 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The a(2) = 1 through a(7) = 12 partitions:
  (2)  (3)   (4)    (5)     (6)      (7)
       (21)  (31)   (32)    (42)     (43)
             (211)  (41)    (51)     (52)
                    (311)   (222)    (61)
                    (2111)  (321)    (322)
                            (411)    (421)
                            (3111)   (511)
                            (21111)  (2221)
                                     (3211)
                                     (4111)
                                     (31111)
                                     (211111)

The strict case is A000009, except that a(0) = a(1) = 0.
Allowing any alternating product >= 1 gives A000041, reverse A344607.
Ranked by A028983 (reverse A347465), which has complement A028982.
The complement is counted by A119620, reverse A347443.
The multiplicative version is A339890, weak A347456, reverse A347705.
The even-length case is A344608.
Allowing any integer reverse-alternating product gives A347445.
Allowing any integer alternating product gives A347446.
The reverse version is A347449, also the odd-length case.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A347461 counts possible alternating products of partitions.
Cf. A000070, A086543, A100824, A236913, A325534, A325535, A339846, A344654, A345196, A347440, A347444, A347462.

a:= n-> (p-> p(n)-p(iquo(n, 2)))(combinat[numbpart]):
seq(a(n), n=0..63);  # Alois P. Heinz, Oct 04 2021

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],altprod[#]>1&]],{n,0,30}]

a(n) = A000041(n) - A119620(n).

A348377 Number of non-alternating compositions of n, excluding twins (x,x).

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 19, 45, 98, 208, 436, 906, 1861, 3803, 7731, 15659, 31628, 63747, 128257, 257722, 517338, 1037652, 2079983, 4167325, 8346203, 16710572, 33449694, 66944254, 133959020, 268028868, 536231902, 1072737537, 2145905284, 4292486690, 8586035992

Offset: 0

Gus Wiseman, Oct 26 2021

nonn

First differs from A348382 at a(6) = 19, A348382(6) = 17. The two non-alternating non-twin compositions of 6 that are not an anti-run are (1,2,3) and (3,2,1).

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 {1..n}.

			The a(3) = 1 through a(6) = 19 compositions:
  (1,1,1)  (1,1,2)    (1,1,3)      (1,1,4)
           (2,1,1)    (1,2,2)      (1,2,3)
           (1,1,1,1)  (2,2,1)      (2,2,2)
                      (3,1,1)      (3,2,1)
                      (1,1,1,2)    (4,1,1)
                      (1,1,2,1)    (1,1,1,3)
                      (1,2,1,1)    (1,1,2,2)
                      (2,1,1,1)    (1,1,3,1)
                      (1,1,1,1,1)  (1,2,2,1)
                                   (1,3,1,1)
                                   (2,1,1,2)
                                   (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

The version for patterns is A000670(n) - A344605(n).
Non-twin compositions are counted by A051049.
The complement is counted by A344604.
An unordered version is A344654.
The complement is ranked by A345167 \/ A007582.
These compositions are ranked by A345168 \ A007582.
Including twins gives A345192, complement A025047.
The version for factorizations is A347706, or A348380 with twins.
The non-anti-run case is A348382.
A001250 counts alternating permutations.
A011782 counts compositions, strict A032020.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A261983 counts non-anti-run compositions, complement A003242.
A325535 counts inseparable partitions, ranked by A335448.
A344614 counts compositions avoiding (1,2,3) and (3,2,1) adjacent.
A345165 = partitions with no alternating permutations, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
Cf. A005649, A178470, A238279, A333755, A335126, A344653, A344740, A345166, A345169, A345173, A348379, A348381.

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

For n > 0, a(n) = A345192(n) - 1 if n is even; otherwise A345192(n).

a(26) onwards from Andrew Howroyd, Jan 31 2024

A347444 Number of odd-length integer partitions of n with integer alternating product.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 4, 8, 7, 14, 13, 24, 21, 40, 35, 62, 55, 99, 85, 151, 128, 224, 195, 331, 283, 481, 416, 690, 593, 980, 844, 1379, 1189, 1918, 1665, 2643, 2292, 3630, 3161, 4920, 4299, 6659, 5833, 8931, 7851, 11905, 10526, 15805, 13987, 20872, 18560, 27398

Offset: 0

Gus Wiseman, Sep 14 2021

nonn

We define the alternating product of a sequence (y_1, ... ,y_k) to be the Product_i y_i^((-1)^(i-1)).

The reverse version (integer reverse-alternating product) is the same.

			The a(1) = 1 through a(9) = 14 partitions:
  (1)  (2)  (3)    (4)    (5)      (6)      (7)        (8)        (9)
            (111)  (211)  (221)    (222)    (322)      (332)      (333)
                          (311)    (411)    (331)      (422)      (441)
                          (11111)  (21111)  (421)      (611)      (522)
                                            (511)      (22211)    (621)
                                            (22111)    (41111)    (711)
                                            (31111)    (2111111)  (22221)
                                            (1111111)             (32211)
                                                                  (33111)
                                                                  (42111)
                                                                  (51111)
                                                                  (2211111)
                                                                  (3111111)
                                                                  (111111111)

The reciprocal version is A035363.
Allowing any alternating product gives A027193.
The multiplicative version (factorizations) is A347441.
Allowing any length gives A347446, reverse A347445.
Allowing any length and alternating product > 1 gives A347448.
Allowing any reverse-alternating product > 1 gives A347449.
Ranked by A347453.
The even-length instead of odd-length version is A347704.
A000041 counts partitions.
A000302 counts odd-length compositions, ranked by A053738.
A025047 counts wiggly compositions.
A026424 lists numbers with odd bigomega.
A027187 counts partitions of even length, strict A067661.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A339890 counts odd-length factorizations.
A347437 counts factorizations with integer alternating product.
A347461 counts possible alternating products of partitions.
Cf. A000070, A236559, A236913, A236914, A304620, A344654, A347439, A347442, A347456, A347457, A347460, A347462, A347463.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,0,30}]

A344649 Triangle read by rows where T(n,k) is the number of strict integer partitions of 2n with reverse-alternating sum 2k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 1, 3, 3, 2, 1, 0, 1, 0, 1, 4, 4, 3, 2, 1, 0, 1, 0, 1, 5, 6, 4, 3, 2, 1, 0, 1, 0, 1, 7, 7, 6, 4, 3, 2, 1, 0, 1, 0, 1, 8, 10, 8, 6, 4, 3, 2, 1, 0, 1

Offset: 0

Gus Wiseman, Jun 05 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)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. So T(n,k) is the number of strict integer partitions of 2n into an odd number of parts whose conjugate has exactly 2k odd parts.

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

			Triangle begins:
   1
   0   1
   0   0   1
   0   1   0   1
   0   1   1   0   1
   0   1   2   1   0   1
   0   1   3   2   1   0   1
   0   1   3   3   2   1   0   1
   0   1   4   4   3   2   1   0   1
   0   1   5   6   4   3   2   1   0   1
   0   1   7   7   6   4   3   2   1   0   1
   0   1   8  10   8   6   4   3   2   1   0   1
   0   1  10  13  12   8   6   4   3   2   1   0   1
   0   1  11  18  15  12   8   6   4   3   2   1   0   1
   0   1  14  22  21  16  12   8   6   4   3   2   1   0   1
   0   1  15  29  27  23  16  12   8   6   4   3   2   1   0   1
Row n = 8 counts the following partitions (empty columns indicated by dots):
  .  (8,7,1)  (7,6,3)      (7,5,4)   (9,4,3)   (11,3,2)  (13,2,1)  .  (16)
              (8,6,2)      (8,5,3)   (10,4,2)  (12,3,1)
              (9,6,1)      (9,5,2)   (11,4,1)
              (6,4,3,2,1)  (10,5,1)
Row n = 9 counts the following partitions (empty columns indicated by dots, A..I = 10..18):
  .  981   873     765     954   B43   D32   F21   .  I
           972     864     A53   C42   E31
           A71     963     B52   D41
           65421   A62     C51
           75321   B61
                   84321

The non-reversed version is A152146.
The non-reversed non-strict version is A239830.
Column k = 2 is A343941.
The non-strict version is A344610.
Row sums are A344650.
Right half of even-indexed rows of A344739.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A067659 counts strict partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
A124754 gives alternating sum of standard compositions (reverse: A344618).
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.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A344741 counts partitions of 2n with reverse-alternating sum -2.
Cf. A000070, A000097, A003242, A006330, A027187, A114121, A116406, A239829, A344607, A344608, A344651, A344654.

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

A345196 Number of integer partitions of n with reverse-alternating sum equal to the reverse-alternating sum of their conjugate.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 3, 4, 4, 4, 8, 11, 11, 11, 20, 27, 29, 31, 48, 65, 70, 74, 109, 145, 160, 172, 238, 314, 345, 372, 500, 649, 721, 782, 1019, 1307, 1451, 1577, 2015, 2552, 2841, 3098, 3885, 4867, 5418, 5914, 7318, 9071, 10109, 11050

Offset: 0

Gus Wiseman, Jun 26 2021

nonn

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. By conjugation, this is also (-1)^(r-1) times the number of odd parts, where r is the greatest part. So a(n) is the number of integer partitions of n of even rank with the same number of odd parts as their conjugate.

			The a(5) = 1 through a(12) = 11 partitions:
  (311)  (321)  (43)    (44)    (333)    (541)    (65)      (66)
                (2221)  (332)   (531)    (4321)   (4322)    (552)
                (4111)  (2222)  (32211)  (32221)  (4331)    (4332)
                        (4211)  (51111)  (52111)  (4421)    (4422)
                                                  (6311)    (4431)
                                                  (222221)  (6411)
                                                  (422111)  (33222)
                                                  (611111)  (53211)
                                                            (222222)
                                                            (422211)
                                                            (621111)

The non-reverse version is A277103.
Comparing even parts to odd conjugate parts gives A277579.
Comparing signs only gives A340601.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
A124754 gives alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A000070, A000097, A006330, A027187, A027193, A236559, A239829, A257991, A344607, A344608, A344651, A344654.

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

A344743 Number of integer partitions of 2n with reverse-alternating sum < 0.

Original entry on oeis.org

0, 0, 1, 3, 7, 15, 29, 54, 96, 165, 275, 449, 716, 1123, 1732, 2635, 3955, 5871, 8620, 12536, 18065, 25821, 36617, 51560, 72105, 100204, 138417, 190134, 259772, 353134, 477734, 643354, 862604, 1151773, 1531738, 2029305, 2678650, 3523378, 4618835, 6035240, 7861292

Offset: 0

Gus Wiseman, Jun 09 2021

nonn

Conjecture: a(n) >= A236914.
The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. So a(n) is the number of even-length partitions of 2n with at least one odd conjugate part. By conjugation, this is also the number of partitions of 2n with greatest part even and at least one odd part.
The alternating sum of a partition is never < 0, so the non-reverse version is A000004.

			The a(2) = 1 through a(5) = 15 partitions:
  (31)  (42)    (53)      (64)
        (51)    (62)      (73)
        (3111)  (71)      (82)
                (3221)    (91)
                (4211)    (3331)
                (5111)    (4222)
                (311111)  (4321)
                          (5221)
                          (5311)
                          (6211)
                          (7111)
                          (322111)
                          (421111)
                          (511111)
                          (31111111)

The ordered version (compositions not partitions) appears to be A008549.
The Heinz numbers are A119899 /\ A300061.
Even bisection of A344608.
The complementary partitions of 2n are counted by A344611.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001523 counts unimodal compositions (partial sums: A174439).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
A124754 gives alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000070, A000097, A006330, A027187, A239830, A343941, A344604, A344607, A344650, 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}]

a(n) = A058696(n) - A344611(n).

a(n) = sum of left half of even-indexed rows of A344612.

More terms from Bert Dobbelaere, Jun 12 2021

A348381 Number of inseparable factorizations of n that are not a twin (x*x).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 30 2021

nonn

First differs from A347706 at a(216) = 3, A347706(216) = 4.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A multiset is inseparable if it has no permutation that is an anti-run, meaning there are always adjacent equal parts. Alternatively, a multiset is inseparable if its maximal multiplicity is at most one plus the sum of its remaining multiplicities.

			The a(n) factorizations for n = 96, 192, 384, 576:
  2*2*2*12      3*4*4*4         4*4*4*6           4*4*4*9
  2*2*2*2*6     2*2*2*24        2*2*2*48          2*2*2*72
  2*2*2*2*2*3   2*2*2*2*12      2*2*2*2*24        2*2*2*2*36
                2*2*2*2*2*6     2*2*2*2*3*8       2*2*2*2*4*9
                2*2*2*2*3*4     2*2*2*2*4*6       2*2*2*2*6*6
                2*2*2*2*2*2*3   2*2*2*2*2*12      2*2*2*2*2*18
                                2*2*2*2*2*2*6     2*2*2*2*3*12
                                2*2*2*2*2*3*4     2*2*2*2*2*2*9
                                2*2*2*2*2*2*2*3   2*2*2*2*2*3*6
                                                  2*2*2*2*2*2*3*3

Positions of nonzero terms are A046099.
Partitions not of this type are counted by A325534 - A000035.
Partitions of this type are counted by A325535 - A000035.
Allowing twins gives A333487.
The case without an alternating permutation is A347706, with twins A348380.
The complement is counted by A348383, without twins A335434.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations of sets.
A008480 counts permutations of prime indices, strict A335489.
A025047 counts alternating or wiggly compositions.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A344654 counts non-twin partitions without an alternating permutation.
A348382 counts non-anti-run compositions that are not a twin.
A348611 counts anti-run ordered factorizations.
Cf. A038548, A336102, A344614, A344653, A344740, A347050, A347437, A347438, A347456, A348379, A348609.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],!MatchQ[#,{x_,x_}]&&Select[Permutations[#],!MatchQ[#,{_,x_,x_,_}]&]=={}&]],{n,100}]

a(n > 1) = A333487(n) - A010052(n).

a(2^n) = A325535(n) - 1 for odd n, otherwise A325535(n).

A348383 Number of factorizations of n that are either separable (have an anti-run permutation) or are a twin (x*x).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 9, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 30 2021

nonn

First differs from A347050 at a(216) = 28, A347050(216) = 27.
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.

			The a(216) = 28 factorizations:
  (2*2*2*3*3*3)  (2*2*2*3*9)  (2*2*6*9)   (3*8*9)   (3*72)   (216)
                 (2*2*3*3*6)  (2*3*4*9)   (4*6*9)   (4*54)
                 (2*3*3*3*4)  (2*3*6*6)   (2*2*54)  (6*36)
                              (3*3*4*6)   (2*3*36)  (8*27)
                              (2*2*3*18)  (2*4*27)  (9*24)
                              (2*3*3*12)  (2*6*18)  (12*18)
                                          (2*9*12)  (2*108)
                                          (3*3*24)
                                          (3*4*18)
                                          (3*6*12)
The a(270) = 20 factorizations:
  (2*3*3*3*5)  (2*3*5*9)   (5*6*9)   (3*90)   (270)
               (3*3*5*6)   (2*3*45)  (5*54)
               (2*3*3*15)  (2*5*27)  (6*45)
                           (2*9*15)  (9*30)
                           (3*3*30)  (10*27)
                           (3*5*18)  (15*18)
                           (3*6*15)  (2*135)
                           (3*9*10)

Positions of 1's are 1 and A000040.
Not requiring separability gives A010052 for n > 1.
Positions of 2's are A323644.
Partitions of this type are counted by A325534(n) + A000035(n + 1).
Partitions of this type are ranked by A335433 \/ A001248.
Partitions not of this type are counted by A325535(n) - A000035(n + 1).
Partitions not of this type are ranked by A345193 = A335448 \ A001248.
Not allowing twins gives A335434, complement A333487,
The case with an alternating permutation is A347050, no twins A348379.
The case without an alternating permutation is A347706, no twins A348380.
The complement is counted by A348381.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A003242 counts anti-run compositions, ranked by A333489.
A025047 counts alternating or wiggly compositions.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A119620, A138364, A336103, A344654, A344740, A347437, A347438, A347456, A347458, 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_,_}]&]!={};
Table[Length[Select[facs[n],MatchQ[#,{x_,x_}]||sepQ[#]&]],{n,100}]

a(n > 1) = A335434(n) + A010052(n), where A010052(n) = 1 if n is a perfect square, otherwise 0.

A345193 Heinz numbers of non-twin (x,x) inseparable partitions.

Original entry on oeis.org

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, 272, 288, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351, 352, 368, 375, 376, 384, 400, 405

Offset: 1

Gus Wiseman, Jun 17 2021

nonn

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.

A multiset is separable if it has an anti-run permutation (no adjacent parts equal). This is equivalent to having maximal multiplicity greater than one plus the sum of the remaining multiplicities. For example, the partition (3,2,2,2,1) has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2), so is separable.

			The sequence of terms together with their prime indices begins:
      8: {1,1,1}          112: {1,1,1,1,4}        232: {1,1,1,10}
     16: {1,1,1,1}        125: {3,3,3}            240: {1,1,1,1,2,3}
     24: {1,1,1,2}        128: {1,1,1,1,1,1,1}    243: {2,2,2,2,2}
     27: {2,2,2}          135: {2,2,2,3}          248: {1,1,1,11}
     32: {1,1,1,1,1}      136: {1,1,1,7}          250: {1,3,3,3}
     40: {1,1,1,3}        144: {1,1,1,1,2,2}      256: {1,1,1,1,1,1,1,1}
     48: {1,1,1,1,2}      152: {1,1,1,8}          272: {1,1,1,1,7}
     54: {1,2,2,2}        160: {1,1,1,1,1,3}      288: {1,1,1,1,1,2,2}
     56: {1,1,1,4}        162: {1,2,2,2,2}        296: {1,1,1,12}
     64: {1,1,1,1,1,1}    176: {1,1,1,1,5}        297: {2,2,2,5}
     80: {1,1,1,1,3}      184: {1,1,1,9}          304: {1,1,1,1,8}
     81: {2,2,2,2}        189: {2,2,2,4}          320: {1,1,1,1,1,1,3}
     88: {1,1,1,5}        192: {1,1,1,1,1,1,2}    324: {1,1,2,2,2,2}
     96: {1,1,1,1,1,2}    208: {1,1,1,1,6}        328: {1,1,1,13}
    104: {1,1,1,6}        224: {1,1,1,1,1,4}      336: {1,1,1,1,2,4}

A000041 counts integer partitions.
A001248 lists Heinz numbers of twins (x,x).
A001250 counts wiggly permutations.
A003242 counts anti-run compositions.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A056239 adds up prime indices,  row sums of A112798.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344740 counts twins and partitions w/ wiggly permutation, rank: A344742.
A345164 counts wiggly permutations of prime indices (with twins: A344606).
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, A316523, A316524, A335126, A344614, A344616, A344652, A344654, A345169, A345173.

Select[Range[100],!MatchQ[FactorInteger[#],{{,2}}]&&Select[Permutations[Flatten[ConstantArray@@@FactorInteger[#]]],!MatchQ[#,{__,x_,x_,_}]&]=={}&]

Complement of A001248 in A335448.

cp's OEIS Frontend

A345162 Number of integer partitions of n with no alternating permutation covering an initial interval of positive integers.

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A347448 Number of integer partitions of n with alternating product > 1.

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A348377 Number of non-alternating compositions of n, excluding twins (x,x).

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A347444 Number of odd-length integer partitions of n with integer alternating product.

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A344649 Triangle read by rows where T(n,k) is the number of strict integer partitions of 2n with reverse-alternating sum 2k.

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A345196 Number of integer partitions of n with reverse-alternating sum equal to the reverse-alternating sum of their conjugate.

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A344743 Number of integer partitions of 2n with reverse-alternating sum < 0.

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A348381 Number of inseparable factorizations of n that are not a twin (x*x).

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