A349055 -id:A349055 - cp's OEIS frontend

A345194 Number of alternating patterns of length n.

1, 1, 2, 6, 22, 102, 562, 3618, 26586, 219798, 2018686, 20393790, 224750298, 2683250082, 34498833434, 475237879950, 6983085189454, 109021986683046, 1802213242949602, 31447143854808378, 577609702827987882, 11139837273501641502, 225075546284489412854

Offset: 0

Gus Wiseman, Jun 17 2021

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
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). An alternating pattern is necessarily an anti-run (A005649).
The version with twins (A344605) is identical to this sequence except with a(2) = 3 instead of 2.
From Gus Wiseman, Jan 16 2022: (Start)
Conjecture: Also the number of weakly up/down patterns of length n, where a sequence is weakly up/down if it is alternately weakly increasing and weakly decreasing, starting with an increase. For example, the a(0) = 1 through a(3) = 6 weakly up/down patterns are:
  ()  (1)  (1,1)  (1,1,1)
           (2,1)  (1,1,2)
                  (2,1,1)
                  (2,1,2)
                  (2,1,3)
                  (3,1,2)
(End)

			The a(0) = 1 through a(3) = 6 alternating patterns:
  ()  (1)  (1,2)  (1,2,1)
           (2,1)  (1,3,2)
                  (2,1,2)
                  (2,1,3)
                  (2,3,1)
                  (3,1,2)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The version for permutations is A001250, complement A348615.
The version for compositions is A025047, complement A345192.
The version with twins (x,x) is A344605.
The version for perms of prime indices is A345164, complement A350251.
The version for factorizations is A348610, complement A348613, weak A349059.
The weak version is A349058, complement A350138, compositions A349052.
The complement is counted by A350252.
A000670 = patterns, ranked by A333217.
A003242 = anti-run compositions.
A005649 = anti-run patterns, complement A069321.
A019536 = necklace patterns.
A129852 and A129853 = up/down and down/up compositions.
A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
A345170 = partitions w/ alternating permutation, complement A345165.
A349055 = normal multisets w/ alternating permutation, complement A349050.
Cf. A006330, A049774, A128761, A335456, A335457, A335517, A344614, A344615, A345167, A345168.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
allnorm[n_]:=If[n<=0,{{}},Function[s, Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],wigQ]],{n,0,6}]

F(p,x) = {sum(k=0, p, (-1)^((k+1)\2)*binomial((p+k)\2, k)*x^k)}
R(n,k) = {Vec(if(k==1, x, 2*F(k-2,-x)/F(k-1,x)-2-(k-2)*x) + O(x*x^n))}
seq(n)= {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Feb 04 2022

a(n) = 2*A350354(n) for n >= 2. - Andrew Howroyd, Feb 04 2022

a(10)-a(18) from Alois P. Heinz, Dec 10 2021

Terms a(19) and beyond from Andrew Howroyd, Feb 04 2022

A349050 Number of multisets of size n that have no alternating permutations and cover an initial interval of positive integers.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 12, 20, 32, 48, 80, 112, 192, 256, 448, 576, 1024, 1280, 2304, 2816, 5120, 6144, 11264, 13312, 24576, 28672, 53248, 61440, 114688, 131072, 245760, 278528, 524288, 589824, 1114112, 1245184, 2359296, 2621440, 4980736, 5505024

Offset: 0

Gus Wiseman, Dec 12 2021

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). Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.

			The multiset {1,2,2,2,2,3,3} has no alternating permutations, even though it does have the three anti-run permutations (2,1,2,3,2,3,2), (2,3,2,1,2,3,2), (2,3,2,3,2,1,2), so is counted under a(7).
The a(2) = 1 through a(7) = 12 multisets:
  {11}  {111}  {1111}  {11111}  {111111}  {1111111}
               {1112}  {11112}  {111112}  {1111112}
               {1222}  {12222}  {111122}  {1111122}
                       {12223}  {111123}  {1111123}
                                {112222}  {1122222}
                                {122222}  {1122223}
                                {122223}  {1222222}
                                {123333}  {1222223}
                                          {1222233}
                                          {1222234}
                                          {1233333}
                                          {1233334}
As compositions:
  (2)  (3)  (4)    (5)      (6)      (7)
            (1,3)  (1,4)    (1,5)    (1,6)
            (3,1)  (4,1)    (2,4)    (2,5)
                   (1,3,1)  (4,2)    (5,2)
                            (5,1)    (6,1)
                            (1,1,4)  (1,1,5)
                            (1,4,1)  (1,4,2)
                            (4,1,1)  (1,5,1)
                                     (2,4,1)
                                     (5,1,1)
                                     (1,1,4,1)
                                     (1,4,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Frankie Mennicucci, On the Number of Strong Dominating Sets in a Graph, Master's Thesis, Montclair State Univ. (2024). See p. 32.
Wikipedia, Alternating permutation

The case of weakly decreasing multiplicities is A025065.
The inseparable case is A336102.
A separable instead of alternating version is A336103.
The version for partitions is A345165.
The version for factorizations is A348380, complement A348379.
The complement (still covering an initial interval) is counted by A349055.
A000670 counts sequences covering an initial interval, anti-run A005649.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions, ranked by A333489.
A025047 = alternating compositions, ranked by A345167, also A025048/A025049.
A049774 counts permutations avoiding the consecutive pattern (1,2,3).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A344654 counts partitions w/o an alternating permutation, ranked by A344653.
Cf. A019472, A052841, A096441, A106351, A106356, A335125, A335126, A344614, A347050, A347706, A348377, A349054.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[allnorm[n],Select[Permutations[#],wigQ]=={}&]],{n,0,7}]

a(n) = if(n==0, 0, if(n%2==0, (n+2)*2^(n/2-3), (n-1)*2^((n-1)/2-2))) \\ Andrew Howroyd, Jan 13 2024

a(n) = A011782(n) - A349055(n).

a(n) = (n+2)*2^(n/2-3) for even n > 0; a(n) = (n-1)*2^((n-5)/2) for odd n. - Andrew Howroyd, Jan 13 2024

Terms a(10) and beyond from Andrew Howroyd, Jan 13 2024

A069321 Stirling transform of A001563: a(0) = 1 and a(n) = Sum_{k=1..n} Stirling2(n,k)kk! for n >= 1.

Original entry on oeis.org

1, 1, 5, 31, 233, 2071, 21305, 249271, 3270713, 47580151, 760192505, 13234467511, 249383390393, 5057242311031, 109820924003705, 2542685745501751, 62527556173577273, 1627581948113854711, 44708026328035782905, 1292443104462527895991, 39223568601129844839353

Offset: 0

Karol A. Penson, Mar 14 2002

nonn

The number of compatible bipartitions of a set of cardinality n for which at least one subset is not underlined. E.g., for n=2 there are 5 such bipartitions: {1 2}, {1}{2}, {2}{1}, {1}{2}, {2}{1}. A005649 is the number of bipartitions of a set of cardinality n. A000670 is the number of bipartitions of a set of cardinality n with none of the subsets underlined. - Kyle Petersen, Mar 31 2005
a(n) is the cardinality of the image set summed over "all surjections". All surjections means: onto functions f:{1, 2, ..., n} -> {1, 2, ..., k} for every k, 1 <= k <= n.  a(n) = Sum_{k=1..n} A019538(n, k)*k. - Geoffrey Critzer, Nov 12 2012
From Gus Wiseman, Jan 15 2022: (Start)
For n > 1, also the number of finite sequences of length n + 1 covering an initial interval of positive integers with at least two adjacent equal parts, or non-anti-run patterns, ranked by the intersection of A348612 and A333217. The complement is counted by A005649. For example, the a(3) = 31 patterns, grouped by sum, are:
  (1111)  (1222)  (1122)  (1112)  (1233)  (1223)
          (2122)  (1221)  (1121)  (1332)  (1322)
          (2212)  (2112)  (1211)  (2133)  (2213)
          (2221)  (2211)  (2111)  (2331)  (2231)
          (1123)                  (3312)  (3122)
          (1132)                  (3321)  (3221)
          (2113)
          (2311)
          (3112)
          (3211)
Also the number of ordered set partitions of {1,...,n + 1} with two successive vertices together in some block.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..200
Benoit Cloitre, On the fractal behavior of primes, 2011. [internet archive]
Benoit Cloitre, On the fractal behavior of primes, 2011.
D. Foata and D. Zeilberger, The Graphical Major Index, arXiv:math/9406220 [math.CO], 1994.
D. Foata and D. Zeilberger, Graphical major indices, J. Comput. Appl. Math. 68 (1996), no. 1-2, 79-101.

Cf. A000629, A001563, A232472.
The complement is counted by A005649.
A version for permutations of prime indices is A336107.
A version for factorizations is A348616.
Dominated (n > 1) by A350252, complement A345194, compositions A345192.
A000670 = patterns, ranked by A333217.
A001250 = alternating permutations, complement A348615.
A003242 = anti-run compositions, ranked by A333489.
A019536 = necklace patterns.
A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
A261983 = not-anti-run compositions, ranked by A348612.
A333381 = anti-runs of standard compositions.
Cf. A000110, A008277, A055932, A106356, A238424, A333382, A335456, A336103, A349050, A349055, A350138.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n, j), j=1..n))
    end:
a:= n-> `if`(n=0, 2, b(n+1)-b(n))/2:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 02 2018

max = 20; t = Sum[n^(n - 1)x^n/n!, {n, 1, max}]; Range[0, max]!CoefficientList[Series[D[1/(1 - y(Exp[x] - 1)), y] /. y -> 1, {x, 0, max}], x] (* Geoffrey Critzer, Nov 12 2012 *)
Prepend[Table[Sum[StirlingS2[n, k]*k*k!, {k, n}], {n, 18}], 1] (* Michael De Vlieger, Jan 03 2016 *)
a[n_] := (PolyLog[-n-1, 1/2] - PolyLog[-n, 1/2])/4; a[0] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 30 2016 *)
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],MemberQ[Differences[#],0]&]],{n,0,8}] (* Gus Wiseman, Jan 15 2022 *)

{a(n)=polcoeff(1+sum(m=1, n, (2*m-1)!/(m-1)!*x^m/prod(k=1, m, 1+(m+k-1)*x+x*O(x^n))), n)} \\ Paul D. Hanna, Oct 28 2013

Representation as an infinite series: a(0) = 1 and a(n) = Sum_{k>=2} (k^n*(k-1)/(2^k))/4 for n >= 1. This is a Dobinski-type summation formula.
E.g.f.: (exp(x) - 1)/((2 - exp(x))^2).
a(n) = (1/2)*(A000670(n+1) - A000670(n)).
O.g.f.: 1 + Sum_{n >= 1} (2*n-1)!/(n-1)! * x^n / (Product_{k=1..n} (1 + (n + k - 1)*x)). - Paul D. Hanna, Oct 28 2013
a(n) = (A000629(n+1) - A000629(n))/4. - Benoit Cloitre, Oct 20 2002
a(n) = A232472(n-1)/2. - Vincenzo Librandi, Jan 03 2016
a(n) ~ n! * n / (4 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018
a(n > 0) = A000607(n + 1) - A005649(n). - Gus Wiseman, Jan 15 2022

A349058 Number of weakly alternating patterns of length n.

Original entry on oeis.org

1, 1, 3, 11, 43, 203, 1123, 7235, 53171, 439595, 4037371, 40787579, 449500595, 5366500163, 68997666867, 950475759899, 13966170378907, 218043973366091, 3604426485899203, 62894287709616755, 1155219405655975763, 22279674547003283003, 450151092568978825707

Offset: 0

Gus Wiseman, Dec 04 2021

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

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

			The a(1) = 1 through a(3) = 11 patterns:
  (1)  (1,1)  (1,1,1)
       (1,2)  (1,1,2)
       (2,1)  (1,2,1)
              (1,2,2)
              (1,3,2)
              (2,1,1)
              (2,1,2)
              (2,1,3)
              (2,2,1)
              (2,3,1)
              (3,1,2)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The strict case is A001250, complement A348615.
The strong case of compositions is A025047, ranked by A345167.
The unordered version is A052955.
The strong case is A345194, with twins A344605. Also the directed case.
The version for compositions is A349052, complement A349053.
The version for permutations of prime indices: A349056, complement A349797.
The version for compositions is ranked by A349057.
The version for ordered factorizations is A349059, strong A348610.
The version for partitions is A349060, complement A349061.
A003242 counts Carlitz (anti-run) compositions.
A005649 counts anti-run patterns.
A344604 counts alternating compositions with twins.
A345163 counts normal partitions with an alternating permutation.
A345170 counts partitions w/ an alternating permutation, complement A345165.
A345192 counts non-alternating compositions, ranked by A345168.
A349055 counts multisets w/ an alternating permutation, complement A349050.
Cf. A049774, A096441, A129852, A129853, A336103, A344614, A344615, A344740, A345164, A348613, A349794.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s, y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+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/@allnorm[n],whkQ[#]||whkQ[-#]&]],{n,0,6}]

R(n,k)={my(v=vector(k,i,1), u=vector(n)); for(r=1, n, if(r%2==0, my(s=v[k]); forstep(i=k, 2, -1, v[i] = s - v[i-1]); v[1] = s); for(i=2, k, v[i] += v[i-1]); u[r]=v[k]); u}
seq(n)= {concat([1], -vector(n,i,1) + 2*sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ) )} \\ Andrew Howroyd, Jan 13 2024

a(9)-a(18) from Alois P. Heinz, Dec 10 2021

a(19) onwards from Andrew Howroyd, Jan 13 2024

A350252 Number of non-alternating patterns of length n.

Original entry on oeis.org

0, 0, 1, 7, 53, 439, 4121, 43675, 519249, 6867463, 100228877, 1602238783, 27866817297, 524175098299, 10606844137009, 229807953097903, 5308671596791901, 130261745042452855, 3383732450013895721, 92770140175473602755, 2677110186541556215233

Offset: 0

Gus Wiseman, Jan 13 2022

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
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). An alternating pattern is necessarily an anti-run (A005649).
Conjecture: Also the number of non-weakly up/down (or down/up) patterns of length n. For example:
- The a(3) = 7 non-weakly up/down patterns:
  (121), (122), (123), (132), (221), (231), (321)
- The a(3) = 7 non-weakly down/up patterns:
  (112), (123), (211), (212), (213), (312), (321)
- The a(3) = 7 non-alternating patterns (see example for more):
  (111), (112), (122), (123), (211), (221), (321)

			The a(2) = 1 and a(3) = 7 non-alternating patterns:
  (1,1)  (1,1,1)
         (1,1,2)
         (1,2,2)
         (1,2,3)
         (2,1,1)
         (2,2,1)
         (3,2,1)
The a(4) = 53 non-alternating patterns:
  2112   3124   4123   1112   2134   1234   3112   2113   1123
  2211   3214   4213   1211   2314   1243   3123   2123   1213
  2212   3412   4312   1212   2341   1324   3211   2213   1223
         3421   4321   1221   2413   1342   3212   2311   1231
                       1222   2431   1423   3213   2312   1232
                                     1432   3312   2313   1233
                                            3321   2321   1312
                                                   2331   1321
                                                          1322
                                                          1323
                                                          1332

Andrew Howroyd, Table of n, a(n) for n = 0..200

The unordered version is A122746.
The version for compositions is A345192, ranked by A345168, weak A349053.
The complement is counted by A345194, weak A349058.
The version for factorizations is A348613, complement A348610, weak A350139.
The strict case (permutations) is A348615, complement A001250.
The weak version for partitions is A349061, complement A349060.
The weak version for perms of prime indices is A349797, complement A349056.
The weak version is A350138.
The version for perms of prime indices is A350251, complement A345164.
A000670 = patterns (ranked by A333217).
A003242 = anti-run compositions, complement A261983, ranked by A333489.
A005649 = anti-run patterns, complement A069321.
A019536 = necklace patterns.
A025047/A129852/A129853 = alternating compositions, ranked by A345167.
A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
A345163 = normal partitions w/ alternating permutation, complement A345162.
A345170 = partitions w/ alternating permutation, complement A345165.
A349055 = normal multisets w/ alternating permutation, complement A349050.
Cf. A049774, A052955, A096441, A128761, A274230, A335456, A335457, A336103, A344605, A344614.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&& Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Join@@Permutations/@allnorm[n],!wigQ[#]&]],{n,0,6}]

a(n) = A000670(n) - A345194(n).

Terms a(9) and beyond from Andrew Howroyd, Feb 04 2022

A350138 Number of non-weakly alternating patterns of length n.

Original entry on oeis.org

0, 0, 0, 2, 32, 338, 3560, 40058, 492664, 6647666, 98210192, 1581844994, 27642067000, 521491848218, 10572345303576, 229332715217954, 5301688511602448, 130152723055769810, 3381930236770946120, 92738693031618794378, 2676532576838728227352

Offset: 0

Gus Wiseman, Dec 24 2021

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
Conjecture: The directed cases, which count non-weakly up/down or non-weakly down/up patterns, are both equal to the strong case: A350252.

			The a(4) = 32 patterns:
  (1,1,2,3)  (2,1,1,2)  (3,1,1,2)  (4,1,2,3)
  (1,2,2,1)  (2,1,1,3)  (3,1,2,3)  (4,2,1,3)
  (1,2,3,1)  (2,1,2,3)  (3,1,2,4)  (4,3,1,2)
  (1,2,3,2)  (2,1,3,4)  (3,2,1,1)  (4,3,2,1)
  (1,2,3,3)  (2,3,2,1)  (3,2,1,2)
  (1,2,3,4)  (2,3,3,1)  (3,2,1,3)
  (1,2,4,3)  (2,3,4,1)  (3,2,1,4)
  (1,3,2,1)  (2,4,3,1)  (3,3,2,1)
  (1,3,3,2)             (3,4,2,1)
  (1,3,4,2)
  (1,4,3,2)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The unordered version is A274230, complement A052955.
The strong case of compositions is A345192, ranked by A345168.
The strict case is A348615, complement A001250.
For compositions we have A349053, complement A349052, ranked by A349057.
The complement is counted by A349058.
The version for partitions is A349061, complement A349060.
The version for permutations of prime indices: A349797, complement A349056.
The version for ordered factorizations is A350139, complement A349059.
The strong case is A350252, complement A345194. Also the directed case?
A003242 = Carlitz compositions, complement A261983, ranked by A333489.
A005649 = anti-run patterns, complement A069321.
A025047/A129852/A129853 = alternating compositions, ranked by A345167.
A345163 = normal partitions w/ alternating permutation, complement A345162.
A345170 = partitions w/ alternating permutation, complement A345165.
A349055 = normal multisets w/ alternating permutation, complement A349050.
Cf. A049774, A096441, A336103, A344605, A344614, A344615, A344740, A348610, A349794.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+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/@allnorm[n],!whkQ[#]&&!whkQ[-#]&]],{n,0,6}]

R(n,k)={my(v=vector(k,i,1), u=vector(n)); for(r=1, n, if(r%2==0, my(s=v[k]); forstep(i=k, 2, -1, v[i] = s - v[i-1]); v[1] = s); for(i=2, k, v[i] += v[i-1]); u[r]=v[k]); u}
seq(n)= {concat([0], vector(n,i,1) + sum(k=1, n, (vector(n,i,k^i) - 2*R(n, k))*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ) )} \\ Andrew Howroyd, Jan 13 2024

a(n) = A000670(n) - A349058(n).

a(9) onwards from Andrew Howroyd, Jan 13 2024

A356935 Numbers whose prime indices all have odd bigomega (number of prime factors with multiplicity). Products of primes indexed by elements of A026424. MM-numbers of finite multisets of finite odd-length multisets of positive integers.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 19, 25, 27, 31, 33, 37, 41, 45, 51, 55, 57, 59, 61, 67, 71, 75, 81, 83, 85, 93, 95, 99, 103, 107, 109, 111, 113, 121, 123, 125, 127, 131, 135, 153, 155, 157, 165, 171, 177, 179, 181, 183, 185, 187, 191, 193, 197, 201, 205, 209, 211, 213

Offset: 1

Gus Wiseman, Sep 12 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. We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(n). For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The initial terms and corresponding multiset partitions:
   1: {}
   3: {{1}}
   5: {{2}}
   9: {{1},{1}}
  11: {{3}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  31: {{5}}
  33: {{1},{3}}
  37: {{1,1,2}}
  41: {{6}}
  45: {{1},{1},{2}}
  51: {{1},{4}}
  55: {{2},{3}}
  57: {{1},{1,1,1}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Odd-size multisets are ctd by A000302, A027193, A058695, rkd by A026424.
Other types: A050330, A356932, A356933, A356934.
Other conditions: A302478, A302492, A356930, A356939, A356940, A356944, A356945.
Cf. A000040, A000720, A003963, A270995, A302242, A304969, A349050, A349055.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[Times@@Length/@primeMS/@primeMS[#]]&]

A356944 MM-numbers of multisets of gapless multisets of positive integers. Products of primes indexed by elements of A073491.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.
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.
We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(n). For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The initial terms and corresponding multiset partitions:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  10: {{},{2}}
  11: {{3}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  16: {{},{},{},{}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Gapless multisets are counted by A034296, ranked by A073491.
The initial version is A356955.
Other types: A356233, A356941, A356942, A356943.
Other conditions: A302478, A302492, A356930, A356935, A356939, A356940.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A011782 counts multisets covering an initial interval.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A003963, A076610, A270995, A302242, A349050, A349055.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Select[Range[100],And@@nogapQ/@primeMS/@primeMS[#]&]

A356934 Number of multisets of odd-size multisets whose multiset union is a size-n multiset covering an initial interval with weakly decreasing multiplicities.

Original entry on oeis.org

1, 1, 2, 6, 17, 46, 166, 553, 2093

Offset: 0

Gus Wiseman, Sep 09 2022

			The a(1) = 1 through a(4) = 17 multiset partitions:
  {{1}}  {{1},{1}}  {{1,1,1}}      {{1},{1,1,1}}
         {{1},{2}}  {{1,1,2}}      {{1},{1,1,2}}
                    {{1,2,3}}      {{1},{1,2,2}}
                    {{1},{1},{1}}  {{1},{1,2,3}}
                    {{1},{1},{2}}  {{1},{2,3,4}}
                    {{1},{2},{3}}  {{2},{1,1,1}}
                                   {{2},{1,1,2}}
                                   {{2},{1,1,3}}
                                   {{2},{1,3,4}}
                                   {{3},{1,1,2}}
                                   {{3},{1,2,4}}
                                   {{4},{1,2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{1},{2}}
                                   {{1},{1},{2},{2}}
                                   {{1},{1},{2},{3}}
                                   {{1},{2},{3},{4}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Odd-size multisets are counted by A000302, A027193, A058695, ranked by A026424.
Other conditions: A035310, A063834, A330783, A356938, A356943, A356954.
Other types: A050330, A356932, A356933, A356935.
Cf. A055887, A072233, A270995, A304969, A349050, A349055.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n],OddQ[Times@@Length/@#]&]],{n,0,5}]

A356937 Number of multisets of intervals whose multiset union is of size n and covers an initial interval of positive integers.

Original entry on oeis.org

1, 1, 3, 9, 29, 94, 310, 1026, 3411, 11360, 37886, 126442, 422203, 1410189, 4711039, 15740098, 52593430, 175742438, 587266782, 1962469721, 6558071499, 21915580437, 73237274083, 244744474601, 817889464220, 2733235019732, 9133973730633, 30524096110942, 102006076541264

Offset: 0

Gus Wiseman, Sep 08 2022

nonn

An interval such as {3,4,5} is a set with all differences of adjacent elements equal to 1.

			The a(1) = 1 through a(3) = 9 set multipartitions (multisets of sets):
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{1}}  {{1},{1,2}}
         {{1},{2}}  {{1},{2,3}}
                    {{2},{1,2}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Cf. A055887, A063834, A270995, A304969, A349050, A349055.
Intervals are counted by A000012, A001227, ranked by A073485.
Other conditions: A034691, A116540, A255906, A356933, A356942.
Other types: A107742, A356936, A356938, A356939.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
Table[Length[Select[Join@@mps/@allnorm[n],And@@chQ/@#&]],{n,0,5}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
R(n,k) = {EulerT(vector(n, j, max(0, 1+k-j)))}
seq(n) = {my(A=1+O(y*y^n)); for(k = 1, n, A += x^k*(1 + y*Ser(R(n,k), y) - polcoef(1/(1 - x*A) + O(x^(k+2)), k+1))); Vec(subst(A,x,1))} \\ Andrew Howroyd, Jan 01 2023

Terms a(10) and beyond from Andrew Howroyd, Jan 01 2023

cp's OEIS Frontend

A345194 Number of alternating patterns of length n.

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A349050 Number of multisets of size n that have no alternating permutations and cover an initial interval of positive integers.

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A069321 Stirling transform of A001563: a(0) = 1 and a(n) = Sum_{k=1..n} Stirling2(n,k)*k*k! for n >= 1.

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A349058 Number of weakly alternating patterns of length n.

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A350252 Number of non-alternating patterns of length n.

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A350138 Number of non-weakly alternating patterns of length n.

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A356935 Numbers whose prime indices all have odd bigomega (number of prime factors with multiplicity). Products of primes indexed by elements of A026424. MM-numbers of finite multisets of finite odd-length multisets of positive integers.

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A069321 Stirling transform of A001563: a(0) = 1 and a(n) = Sum_{k=1..n} Stirling2(n,k)kk! for n >= 1.