A325534 -id:A325534 - cp's OEIS frontend

A345171 Numbers whose multiset of prime factors has no alternating permutation.

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 80, 81, 88, 96, 104, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 270, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

First differs from A335448 in having 270.
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 has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
Also Heinz numbers of integer partitions without a wiggly permutation, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   32: {1,1,1,1,1}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   49: {4,4}
   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}

Removing squares of primes A001248 gives A344653, counted by A344654.
A superset of A335448, which is counted by A325535.
Positions of 0's in A345164.
The partitions with these Heinz numbers are counted by A345165.
The complement is A345172, counted by A345170.
The separable case is A345173, counted by A345166.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions, complement A261983.
A025047 counts alternating or wiggly compositions, directed A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A344606 counts alternating permutations of prime indices with twins.
A344742 ranks twins and partitions with an alternating permutation.
A345192 counts non-alternating compositions.
Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344604, A344616, A344652, A344740, A345163, A345168, A345193, A345195, A348380, A348609.

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

A277103 Number of partitions of n for which the number of odd parts is equal to the positive alternating sum of the parts.

Original entry on oeis.org

1, 1, 0, 1, 3, 3, 1, 3, 10, 10, 4, 10, 27, 27, 13, 28, 69, 69, 37, 72, 161, 162, 96, 171, 361, 364, 230, 388, 768, 777, 522, 836, 1581, 1605, 1128, 1739, 3145, 3203, 2345, 3495, 6094, 6225, 4712, 6831, 11511, 11794, 9198, 13010, 21293, 21875, 17496, 24239

Offset: 0

Emeric Deutsch, Oct 18 2016

nonn

It follows by conjugation that the partition statistics "alternating sum" and "number of odd parts" are equidistributed. Consequently, the self-conjugate partitions satisfy the required condition.
In the first Maple program (improvable) AS gives the positive alternating sum of a finite sequence s, OP gives the number of odd terms of a finite sequence of positive integers.
For the specified value of n, the second Maple program lists the partitions of n counted by a(n).
Number of integer partitions of n with the same number of odd parts as their conjugate. - Gus Wiseman, Jun 27 2021

			a(3) = 1 because we have [2,1]. The partitions [3] and [1,1,1] do not qualify.
a(4) = 3 because we have [3,1], [2,2], and [2,1,1]. The partitions [4] and [1,1,1,1] do not qualify.

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

Comparing even parts to odd conjugate parts gives A277579.
Comparing product of parts to product of conjugate parts gives A325039.
The reverse version is A345196.
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).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A000070, A000097, A000700, A006330, A027187, A027193, A236559, A257991, A325534, A325535, A344607, A344651.

with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: OP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 1 then ct := ct+1 else  end if end do: ct end proc: a := proc (n) local P, c, k: P := partition(n): c := 0: for k to nops(P) do if AS(P[k]) = OP(P[k]) then c := c+1 else end if end do: c end proc: seq(a(n), n = 0 .. 50);
n := 8: with(combinat): AS := proc (s) options operator, arrow: abs(add((-1)^(i-1)*s[i], i = 1 .. nops(s))) end proc: OP := proc (s) local ct, j: ct := 0: for j to nops(s) do if `mod`(s[j], 2) = 1 then ct := ct+1 else  end if end do: ct end proc: P := partition(n): C := {}: for k to nops(P) do if AS(P[k]) = OP(P[k]) then C := `union`(C, {P[k]}) else  end if end do: C;
# alternative Maple program:
b:= proc(n, i, s, t) option remember; `if`(n=0,
      `if`(s=0, 1, 0), `if`(i<1, 0, b(n, i-1, s, t)+
      `if`(i>n, 0, b(n-i, i, s+t*i-irem(i, 2), -t))))
    end:
a:= n-> b(n$2, 0, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 19 2016

b[n_, i_, s_, t_] := b[n, i, s, t] = If[n == 0, If[s == 0, 1, 0], If[i<1, 0, b[n, i-1, s, t] + If[i>n, 0, b[n-i, i, s + t*i - Mod[i, 2], -t]]]]; a[n_] := b[n, n, 0, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]]; Table[Length[Select[IntegerPartitions[n],Count[#,?OddQ]==Count[conj[#],?OddQ]&]],{n,0,15}] (* Gus Wiseman, Jun 27 2021 *)

A347446 Number of integer partitions of n with integer alternating product.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 31, 37, 54, 62, 84, 100, 134, 157, 207, 241, 314, 363, 463, 537, 685, 785, 985, 1138, 1410, 1616, 1996, 2286, 2801, 3201, 3885, 4434, 5363, 6098, 7323, 8329, 9954, 11293, 13430, 15214, 18022, 20383, 24017, 27141, 31893, 35960

Offset: 0

Gus Wiseman, Sep 15 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(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (41)     (33)      (61)
             (111)  (31)    (221)    (42)      (322)
                    (211)   (311)    (51)      (331)
                    (1111)  (2111)   (222)     (421)
                            (11111)  (411)     (511)
                                     (2211)    (2221)
                                     (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

Allowing any reverse-alternating product >= 1 gives A344607.
Allowing any alternating product <= 1 gives A119620, reverse A347443.
Allowing any reverse-alternating product < 1 gives A344608.
The multiplicative version (factorizations) is A347437, reverse A347442.
The odd-length case is A347444, ranked by A347453.
The reverse version is A347445, ranked by A347454.
Allowing any alternating product > 1 gives A347448, reverse A347449.
Ranked by A347457.
The even-length case is A347704.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A347461 counts possible alternating products of partitions.
Cf. A025047, A067661, A339890, A347450, A344654, A344740, A347439, A347440, A347451, A347463, A347705.

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

A278990 Number of loopless linear chord diagrams with n chords.

Original entry on oeis.org

1, 0, 1, 5, 36, 329, 3655, 47844, 721315, 12310199, 234615096, 4939227215, 113836841041, 2850860253240, 77087063678521, 2238375706930349, 69466733978519340, 2294640596998068569, 80381887628910919255, 2976424482866702081004, 116160936719430292078411

Offset: 0

N. J. A. Sloane, Dec 07 2016

nonn

See the signed version of these numbers, A000806, for much more information about these numbers.
From Gus Wiseman, Feb 27 2019: (Start)
Also the number of 2-uniform set partitions of {1..2n} containing no two successive vertices in the same block. For example, the a(3) = 5 set partitions are:
  {{1,3},{2,5},{4,6}}
  {{1,4},{2,5},{3,6}}
  {{1,4},{2,6},{3,5}}
  {{1,5},{2,4},{3,6}}
  {{1,6},{2,4},{3,5}}
(End)
From Gus Wiseman, Jul 05 2020: (Start)
Also the number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal and where the first i appears before the first j for i < j. For example, the a(3) = 5 permutations are the following.
  (1,2,3,1,2,3)
  (1,2,3,1,3,2)
  (1,2,3,2,1,3)
  (1,2,3,2,3,1)
  (1,2,1,3,2,3)
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..404 (terms 0..200 from Gheorghe Coserea)
Dmitry Efimov, The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomials, arXiv:1904.08651 [math.CO], 2019.
H. Eriksson and A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.
E. Krasko, I. Labutin, and A. Omelchenko, Enumeration of labelled and unlabelled Hamiltonian Cycles in complete k-partite graphs, arXiv:1709.03218 [math.CO], 2017, Table 1.
E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, arXiv:1601.05073 [math.CO], 2016.
E. Krasko and A. Omelchenko, Enumeration of Chord Diagrams without Loops and Parallel Chords, The Electronic Journal of Combinatorics, 24(3) (2017), #P3.43.
Gus Wiseman, The a(4) = 36 loopless linear chord diagrams.
Donovan Young, Counting Bubbles in Linear Chord Diagrams, arXiv:2311.01569 [math.CO], 2023.
Donovan Young, Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams, arXiv:2408.17232 [math.CO], 2024.

Column k=0 of A079267.
Column k=2 of A293157.
Row n=2 of A322013.
Cf. A000110, A000699 (topologically connected 2-uniform), A000806, A001147 (2-uniform), A003436 (cyclical version), A005493, A170941, A190823 (distance 3+ version), A322402, A324011, A324172.
Anti-run compositions are A003242.
Separable partitions are A325534.
Cf. A007716, A292884, A333489.
Other sequences involving the multiset {1,1,2,2,...,n,n}: A001147, A007717, A020555, A094574, A316972.

[n le 2 select 2-n else (2*n-3)*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 26 2023

RecurrenceTable[{a[n]== (2n-1)a[n-1] +a[n-2], a[0]==1, a[1]==0}, a, {n,0,20}] (* Vaclav Kotesovec, Sep 15 2017 *)
FullSimplify[Table[-I*(BesselI[1/2+n,-1] BesselK[3/2,1] - BesselI[3/2,-1] BesselK[1/2+ n,1]), {n,0,20}]] (* Vaclav Kotesovec, Sep 15 2017 *)
Table[(2 n-1)!! Hypergeometric1F1[-n,-2 n,-2], {n,0,20}] (* Eric W. Weisstein, Nov 14 2018 *)
Table[Sqrt[2/Pi]/E ((-1)^n Pi BesselI[1/2+n,1] +BesselK[1/2+n,1]), {n,0,20}] // FunctionExpand // FullSimplify (* Eric W. Weisstein, Nov 14 2018 *)
twouniflin[{}]:={{}};twouniflin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@twouniflin[Complement[set,s]]]/@Table[{i,j},{j,Select[set,#>i+1&]}];
Table[Length[twouniflin[Range[n]]],{n,0,14,2}] (* Gus Wiseman, Feb 27 2019 *)

seq(N) = {
  my(a = vector(N)); a[1] = 0; a[2] = 1;
  for (n = 3, N, a[n] = (2*n-1)*a[n-1] + a[n-2]);
  concat(1, a);
};
seq(20) \\ Gheorghe Coserea, Dec 09 2016

def A278990_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-1+sqrt(1-2*x))/sqrt(1-2*x) ).egf_to_ogf().list()
A278990_list(30) # G. C. Greubel, Sep 26 2023

From Gheorghe Coserea, Dec 09 2016: (Start)
D-finite with recurrence a(n) = (2*n-1)*a(n-1) + a(n-2), with a(0) = 1, a(1) = 0.
E.g.f. y satisfies: 0 = (1-2*x)*y'' - 3*y' - y.
a(n) - a(n-1) = A003436(n) for all n >= 2. (End)
From Vaclav Kotesovec, Sep 15 2017: (Start)
a(n) = sqrt(2)*exp(-1)*(BesselK(1/2 + n, 1)/sqrt(Pi) - i*sqrt(Pi)*BesselI(1/2 + n, -1)), where i is the imaginary unit.
a(n) ~ 2^(n+1/2) * n^n  / exp(n+1). (End)
a(n) = A114938(n)/n! - Gus Wiseman, Jul 05 2020 (from Alexander Burstein's formula at A114938).
From G. C. Greubel, Sep 26 2023: (Start)
a(n) = (-1)^n * (i/e)*Sqrt(2/Pi) * BesselK(n + 1/2, -1).
G.f.: sqrt(Pi/(2*x)) * exp(-(1+x)^2/(2*x)) * Erfi((1+x)/sqrt(2*x)).
E.g.f.: exp(-1 + sqrt(1-2*x))/sqrt(1-2*x). (End)

a(0)=1 prepended by Gheorghe Coserea, Dec 09 2016

A345163 Number of integer partitions of n with an alternating permutation covering an initial interval of positive integers.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 6, 7, 8, 11, 12, 16, 20, 23, 27, 34, 41, 48, 57, 68, 80, 94, 110, 130, 153, 175, 203, 239, 275, 317, 365, 420, 483, 553, 632, 720, 825, 938, 1064, 1211, 1370, 1550, 1755, 1982, 2235, 2517, 2830, 3182, 3576, 4006, 4487, 5027, 5619, 6275, 7007, 7812

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 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).

A partition with k parts is alternating if and only every part has a multiplicity no greater than k/2, except either the smallest or largest part may have a multiplicity of (k+1)/2 when k is odd. - Andrew Howroyd, Jan 31 2024

			The a(3) = 1 through a(12) = 7 partitions:
  21  211  221  321   3211   3221   3321    4321     33221    33321
                2211  22111  22211  32211   33211    43211    43221
                             32111  222111  322111   322211   332211
                                            2221111  332111   432111
                                                     2222111  3222111
                                                     3221111  3321111
                                                              22221111
For example, the partition (3,3,2,1,1,1,1) has the alternating permutations (1,3,1,3,1,2,1), (1,3,1,2,1,3,1), and (1,2,1,3,1,3,1), so is counted under a(12).

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

Not requiring an alternating permutation gives A000670, ranked by A333217.
The complement in covering partitions is counted by A345162.
Not requiring normality gives A345170, ranked by A345172.
A000041 counts integer partitions.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344605 counts alternating patterns with twins.
A345164 counts alternating permutations of prime indices.
A345165 counts partitions without a alternating permutation, ranked by A345171.
A349051 ranks alternating compositions.
Cf. A000070, A103919, A335126, A344604, A344607, A344615, A344653, A344654, A344740, A345166, A345167, A345168.

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}]

\\ See also A345162 for a faster program.
ok(k,p)={my(S=Set(p)); foreach(S, t, my(c=k+#p-2*(1+#select(x->x==t, p))); if(c<0, return(c==-1 && (t==1||t==k)))); 1}
a(n)={sum(k=1, (sqrtint(8*n+1)-1)\2, s=0; forpart(p=n-binomial(k+1,2), s+=ok(k,Vec(p)), k); s)} \\ Andrew Howroyd, Jan 31 2024

The Heinz numbers of these partitions are A333217 /\ A345172.

a(n) = A000009(n) - A345162(n). - Andrew Howroyd, Jan 31 2024

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

A345958 Numbers whose prime indices have reverse-alternating sum 1.

Original entry on oeis.org

2, 6, 8, 15, 18, 24, 32, 35, 50, 54, 60, 72, 77, 96, 98, 128, 135, 140, 143, 150, 162, 200, 216, 221, 240, 242, 288, 294, 308, 315, 323, 338, 375, 384, 392, 437, 450, 486, 512, 540, 560, 572, 578, 600, 648, 667, 693, 722, 726, 735, 800, 864, 875, 882, 884, 899

Offset: 1

Gus Wiseman, Jul 11 2021

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 reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. Of course, the reverse-alternating sum of prime indices is also the alternating sum of reversed prime indices.
Also numbers with exactly one odd conjugate prime index. Conjugate prime indices are listed by A321650, ranked by A122111.

			The initial terms and their prime indices:
   2: {1}
   6: {1,2}
   8: {1,1,1}
  15: {2,3}
  18: {1,2,2}
  24: {1,1,1,2}
  32: {1,1,1,1,1}
  35: {3,4}
  50: {1,3,3}
  54: {1,2,2,2}
  60: {1,1,2,3}
  72: {1,1,1,2,2}
  77: {4,5}
  96: {1,1,1,1,1,2}
  98: {1,4,4}

The k > 0 version is A000037.
These multisets are counted by A000070.
The k = 0 version is A000290, counted by A000041.
The version for unreversed-alternating sum is A001105.
These partitions are counted by A035363.
These are the positions of 1's in A344616.
The k = 2 version is A345961, counted by A120452.
A000984/A345909/A345911 count/rank compositions with alternating sum 1.
A001791/A345910/A345912 count/rank compositions with alternating sum -1.
A088218 counts compositions with alternating sum 0, ranked by A344619.
A025047 counts wiggly compositions.
A027187 counts partitions with reverse-alternating sum <= 0.
A056239 adds up prime indices, row sums of A112798.
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.
A325534 and A325535 count separable and inseparable partitions.
A344606 counts alternating permutations of prime indices.
A344607 counts partitions with reverse-alternating sum >= 0.
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000097, A027193, A034871, A239830, A341446, A344650, A344651, A344743, A345917, A345918, A345920.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[100],sats[primeMS[#]]==1&]

A348379 Number of factorizations of n with an alternating permutation.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 6, 1, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 1, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 3, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 28 2021

nonn

First differs from A335434 at a(216) = 27, A335434(216) = 28. Also differs from A335434 at a(270) = 19, A335434(270) = 20.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
All of the counted factorizations are separable (A335434).
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(270) = 19 factorizations:
  (2*3*3*15)  (2*3*45)  (2*135)  (270)
  (2*3*5*9)   (2*5*27)  (3*90)
  (3*3*5*6)   (2*9*15)  (5*54)
              (3*3*30)  (6*45)
              (3*5*18)  (9*30)
              (3*6*15)  (10*27)
              (3*9*10)  (15*18)
              (5*6*9)

Wikipedia, Alternating permutation

Partitions not of this type are counted by A345165, ranked by A345171.
Partitions of this type are counted by A345170, ranked by A345172.
Twins and partitions of this type are counted by A344740, ranked by A344742.
The case with twins is A347050.
The complement is counted by A348380, without twins A347706.
The ordered version is A348610.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A038548, A056986, A325534, A335452, A347437, A347438, A347439, A347442, A347456, A347463, A348381, A348383, A348609.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[facs[n],Select[Permutations[#],wigQ]!={}&]],{n,100}]

a(2^n) = A345170(n).

A350842 Number of integer partitions of n with no difference -2.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 16, 24, 30, 40, 54, 69, 89, 118, 146, 187, 239, 297, 372, 468, 575, 711, 880, 1075, 1314, 1610, 1947, 2359, 2864, 3438, 4135, 4973, 5936, 7090, 8466, 10044, 11922, 14144, 16698, 19704, 23249, 27306, 32071, 37639, 44019, 51457, 60113

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (211)   (41)     (51)      (52)
                    (1111)  (221)    (222)     (61)
                            (2111)   (321)     (322)
                            (11111)  (411)     (511)
                                     (2211)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (211111)
                                               (1111111)

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

Heinz number rankings are in parentheses below.
The version for no difference 0 is A000009.
The version for subsets of prescribed maximum is A005314.
The version for all differences < -2 is A025157, non-strict A116932.
The version for all differences > -2 is A034296, strict A001227.
The opposite version is A072670.
The version for no difference -1 is A116931 (A319630), strict A003114.
The multiplicative version is A350837 (A350838), strict A350840.
The strict case is A350844.
The complement for quotients is counted by A350846 (A350845).
A000041 = integer partitions.
A027187 = partitions of even length.
A027193 = partitions of odd length (A026424).
A323092 = double-free partitions (A320340), strict A120641.
A325534 = separable partitions (A335433).
A325535 = inseparable partitions (A335448).
A350839 = partitions with a gap and conjugate gap (A350841).
Cf. A000070, A000929, A001511, A003242, A007359, A018819, A040039, A045690, A045691, A101417, A154402, A323093.

Table[Length[Select[IntegerPartitions[n],FreeQ[Differences[#],-2]&]],{n,0,30}]

A335434 Number of separable factorizations of n into factors > 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 6, 1, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 1, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 3, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Jul 03 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

			The a(n) factorizations for n = 2, 6, 16, 12, 30, 24, 36, 48, 60:
  2  6    16     12     30     24     36       48       60
     2*3  2*8    2*6    5*6    3*8    4*9      6*8      2*30
          2*2*4  3*4    2*15   4*6    2*18     2*24     3*20
                 2*2*3  3*10   2*12   3*12     3*16     4*15
                        2*3*5  2*2*6  2*2*9    4*12     5*12
                               2*3*4  2*3*6    2*3*8    6*10
                                      3*3*4    2*4*6    2*5*6
                                      2*2*3*3  3*4*4    3*4*5
                                               2*2*12   2*2*15
                                               2*2*3*4  2*3*10
                                                        2*2*3*5

The version for partitions is A325534.
The inseparable version is A333487.
The version for multisets with prescribed multiplicities is A335127.
Factorizations are A001055.
Anti-run compositions are A003242.
Inseparable partitions are A325535.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Cf. A106351, A292884, A295370, A333628, A333755, A335463, A335125, A335126, A335407, A335457, A335474, A335516, A335838.

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],Select[Permutations[#],!MatchQ[#,{_,x_,x_,_}]&]!={}&]],{n,100}]

A333487(n) + a(n) = A001055(n).

A335127 A multiset whose multiplicities are the prime indices of n is separable.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 77, 80, 81, 84, 90, 96, 98, 99, 100, 105, 108, 110, 112, 120, 121, 125, 126, 128, 132, 135, 140, 143, 144, 147, 150, 154, 160, 162, 165

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The sequence together with the corresponding multisets begins:
   1: {}
   2: {1}
   4: {1,2}
   6: {1,1,2}
   8: {1,2,3}
   9: {1,1,2,2}
  12: {1,1,2,3}
  15: {1,1,1,2,2}
  16: {1,2,3,4}
  18: {1,1,2,2,3}
  20: {1,1,1,2,3}
  24: {1,1,2,3,4}
  25: {1,1,1,2,2,2}
  27: {1,1,2,2,3,3}
  30: {1,1,1,2,2,3}

The complement is A335126.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are A325534.
Inseparable partitions are A325535.
Separable factorizations are A335434.
Inseparable factorizations are A333487.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Patterns contiguously matched by compositions are A335457.
Cf. A056239, A106351, A112798, A114938, A292884, A335489, A335516, A335838.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Select[Permutations[nrmptn[#]],!MatchQ[#,{_,x_,x_,_}]&]!={}&]

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