A349056 -id:A349056 - cp's OEIS frontend

A349053 Number of non-weakly alternating integer compositions of n.

0, 0, 0, 0, 0, 0, 4, 12, 37, 95, 232, 533, 1198, 2613, 5619, 11915, 25011, 52064, 107694, 221558, 453850, 926309, 1884942, 3825968, 7749312, 15667596, 31628516, 63766109, 128415848, 258365323, 519392582, 1043405306, 2094829709, 4203577778, 8431313237, 16904555958

Offset: 0

Gus Wiseman, Dec 16 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is (strongly) alternating iff it is a weakly alternating anti-run.

			The a(6) = 12 compositions:
  (1,1,2,2,1)  (1,1,2,3)  (1,2,4)
  (1,2,1,1,2)  (1,2,3,1)  (4,2,1)
  (1,2,2,1,1)  (1,3,2,1)
  (2,1,1,2,1)  (2,1,1,3)
               (3,1,1,2)
               (3,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000 (terms 0..55 from Martin Ehrenstein)
Wikipedia, Alternating permutation

Complementary directed versions are A129852/A129853, strong A025048/A025049.
The strong version is A345192.
The complement is counted by A349052.
These compositions are ranked by A349057, strong A345168.
The complementary version for patterns is A349058, strong A345194.
The complementary multiplicative version is A349059, strong A348610.
An unordered version (partitions) is A349061, complement A349060.
The version for ordered prime factorizations is A349797, complement A349056.
The version for patterns is A350138, strong A350252.
The version for ordered factorizations is A350139.
A001250 counts alternating permutations, complement A348615.
A001700 counts compositions of 2n with alternating sum 0.
A003242 counts Carlitz (anti-run) compositions.
A011782 counts compositions, unordered A000041.
A025047 counts alternating compositions, ranked by A345167.
A106356 counts compositions by number of maximal anti-runs.
A344604 counts alternating compositions with twins.
A345164 counts alternating ordered prime factorizations.
A349054 counts strict alternating compositions.
Cf. A102726, A114901, A128761, A261983, A333213, A333755, A344614, A344615, A345165, A345170, A345195, A349799, A349800, A350251, A350252.

wwkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}]||And@@Table[If[EvenQ[m],y[[m]]>=y[[m+1]],y[[m]]<=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!wwkQ[#]&]],{n,0,10}]

a(n) = A011782(n) - A349052(n).

a(21)-a(35) from Martin Ehrenstein, Jan 08 2022

A349052 Number of weakly alternating compositions of n.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 28, 52, 91, 161, 280, 491, 850, 1483, 2573, 4469, 7757, 13472, 23378, 40586, 70438, 122267, 212210, 368336, 639296, 1109620, 1925916, 3342755, 5801880, 10070133, 17478330, 30336518, 52653939, 91389518, 158621355, 275313226, 477850887, 829388075

Offset: 0

Gus Wiseman, Nov 29 2021

nonn

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

			The a(5) = 16 compositions:
  (1,1,1,1,1)  (1,1,1,2)  (1,1,3)  (1,4)  (5)
               (1,1,2,1)  (1,2,2)  (2,3)
               (1,2,1,1)  (1,3,1)  (3,2)
               (2,1,1,1)  (2,1,2)  (4,1)
                          (2,2,1)
                          (3,1,1)
The a(6) = 28 compositions:
  (111111)  (11112)  (1113)  (114)  (15)  (6)
            (11121)  (1122)  (132)  (24)
            (11211)  (1131)  (141)  (33)
            (12111)  (1212)  (213)  (42)
            (21111)  (1311)  (222)  (51)
                     (2121)  (231)
                     (2211)  (312)
                     (3111)  (411)

Andrew Howroyd, Table of n, a(n) for n = 0..1000 (Terms 0..55 from Martin Ehrenstein)

The strong case is A025047, ranked by A345167.
The directed versions are A129852 and A129853, strong A025048 and A025049.
The complement is counted by A349053, strong A345192.
The version for permutations of prime indices is A349056, strong A345164.
The complement is ranked by A349057, strong A345168.
The version for patterns is A349058, strong A345194.
The multiplicative version is A349059, strong A348610.
An unordered version (partitions) is A349060, complement A349061.
The non-alternating case is A349800, ranked by A349799.
A001250 counts alternating permutations, complement A348615.
A001700 counts compositions of 2n with alternating sum 0.
A003242 counts Carlitz (anti-run) compositions.
A011782 counts compositions.
A106356 counts compositions by number of maximal anti-runs.
A344604 counts alternating compositions with twins.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349054 counts strict alternating compositions.
Cf. A000041, A008965, A102726, A114901, A128761, A261983, A333213, A333755, A344614, A344615, A345165, A345195.

whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],whkQ[#]||whkQ[-#]&]],{n,0,10}]

C(n,f)={my(M=matrix(n,n,j,k,k>=j), s=M[,n]); for(b=1, n, f=!f; M=matrix(n,n,j,k, if(k1,M[j-k,k-1]) ))); for(k=2, n, M[,k]+=M[,k-1]); s+=M[,n]); s~}
seq(n) = concat([1], C(n,0) + C(n,1) - vector(n,j,numdiv(j))) \\ Andrew Howroyd, Jan 31 2024

a(21)-a(37) from Martin Ehrenstein, Jan 08 2022

A349057 Numbers k such that the k-th composition in standard order is not weakly alternating.

Original entry on oeis.org

37, 46, 52, 53, 69, 75, 78, 92, 93, 101, 104, 105, 107, 110, 116, 117, 133, 137, 139, 142, 150, 151, 156, 157, 165, 174, 180, 181, 184, 185, 186, 187, 190, 197, 200, 201, 203, 206, 208, 209, 210, 211, 214, 215, 220, 221, 229, 232, 233, 235, 238, 244, 245, 261

Offset: 1

Gus Wiseman, Dec 04 2021

nonn

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

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms and corresponding compositions begin:
   37: (3,2,1)
   46: (2,1,1,2)
   52: (1,2,3)
   53: (1,2,2,1)
   69: (4,2,1)
   75: (3,2,1,1)
   78: (3,1,1,2)
   92: (2,1,1,3)
   93: (2,1,1,2,1)
  101: (1,3,2,1)
  104: (1,2,4)
  105: (1,2,3,1)
  107: (1,2,2,1,1)
  110: (1,2,1,1,2)
  116: (1,1,2,3)
  117: (1,1,2,2,1)

The strong case is A345168, complement A345167, counted by A345192.
The strong anti-run case is A345169, counted by A345195.
Including all non-anti-runs gives A348612, complement A333489.
These compositions are counted by A349053, complement A349052.
The directed cases are counted by A129852 (incr.) and A129853 (decr.).
The complement for patterns is A349058, strong A345194.
The complement for ordered factorizations is A349059, strong A348610.
Partitions of this type are counted by A349061, complement A349060.
Partitions of this type are ranked by A349794.
Non-strict partitions of this type are counted by A349796.
Permutations of prime indices of this type are counted by A349797.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions, complement A261983.
A011782 counts compositions.
A025047 counts alternating/wiggly compositions, directed A025048, A025049.
A345164 counts alternating permutations of prime indices, weak A349056.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349054 counts strict alternating compositions.
Cf. A001700, A096441, A128761, A344615, A344654, A345173, A348613, A349051, A349794, A349795, A349799.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Select[Range[0,100],!whkQ[stc[#]]&&!whkQ[-stc[#]]&]

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Dec 06 2021

nonn

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

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

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

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

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

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

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

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Dec 06 2021

nonn

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

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

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

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

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

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

Original entry on oeis.org

0, 0, 1, 1, 4, 9, 16, 33, 62, 113, 205, 373, 664, 1190, 2113, 3744, 6618, 11683, 20564, 36164, 63489, 111343, 195042, 341357, 596892, 1042976, 1821179, 3178145, 5543173, 9663545, 16839321, 29332231, 51075576, 88908912, 154722756, 269186074, 468221264

Offset: 0

Gus Wiseman, Dec 16 2021

nonn

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

This sequence counts compositions that are weakly but not strongly alternating; also weakly alternating non-anti-run compositions.

			The a(2) = 1 through a(6) = 16 compositions:
  (1,1)  (1,1,1)  (2,2)      (1,1,3)      (3,3)
                  (1,1,2)    (1,2,2)      (1,1,4)
                  (2,1,1)    (2,2,1)      (2,2,2)
                  (1,1,1,1)  (3,1,1)      (4,1,1)
                             (1,1,1,2)    (1,1,1,3)
                             (1,1,2,1)    (1,1,2,2)
                             (1,2,1,1)    (1,1,3,1)
                             (2,1,1,1)    (1,3,1,1)
                             (1,1,1,1,1)  (2,2,1,1)
                                          (3,1,1,1)
                                          (1,1,1,1,2)
                                          (1,1,1,2,1)
                                          (1,1,2,1,1)
                                          (1,2,1,1,1)
                                          (2,1,1,1,1)
                                          (1,1,1,1,1,1)

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

This is the weakly alternating case of A345192, ranked by A345168.
The case of partitions is A349795, ranked by A350137.
The version counting permutations of prime indices is A349798.
These compositions are ranked by A349799.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions, ranked by A333489.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A261983 = non-anti-run compositions, ranked by A348612.
A345165 = partitions without an alternating permutation, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
A345173 = non-alternating anti-run partitions, ranked by A345166.
A345195 = non-alternating anti-run compositions, ranked by A345169.
A348377 = non-alternating non-twin compositions.
A349801 = non-alternating partitions, ranked by A289553.
Weakly alternating:
- A349052 = compositions, directed A129852/A129853, complement A349053.
- A349056 = permutations of prime indices, complement A349797.
- A349057 = complement of standard composition numbers (too dense).
- A349058 = patterns, complement A350138.
- A349059 = ordered factorizations, complement A350139.
- A349060 = partitions, complement A349061.
Cf. A008965, A011782, A027383, A096441, A274230, A333213, A344614, A344615, A348382, A348613, A349796, A350140.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y] &&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],(whkQ[#]||whkQ[-#])&&!wigQ[#]&]],{n,0,10}]

a(n) = A349052(n) - A025047(n). - Andrew Howroyd, Jan 31 2024

a(21) onwards from Andrew Howroyd, Jan 31 2024

A349797 Number of non-weakly alternating permutations of the multiset of prime factors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0

Offset: 1

Gus Wiseman, Dec 24 2021

nonn

First differs from 2 * A326291 at a(90) = 4, A326291(90) = 3.
The first odd term is a(144) = 7, whose non-weakly alternating permutations are shown in the example below.
We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is alternating in the sense of A025047 iff it is a weakly alternating anti-run.
For n > 1, the multiset of prime factors of n is row n of A027746. The prime indices A112798 can also be used.

			The following are the weakly alternating permutations for selected n.
n = 30    60     72      120     144      180
   ---------------------------------------------
    235   2235   22332   22235   222332   22353
    532   2352   23223   22352   223223   23235
          2532   23322   22532   223322   23325
          3225   32232   23225   232232   23523
          5223           23522   233222   23532
          5322           25223   322223   25323
                         25322   322322   32235
                         32252            32253
                         52232            32352
                         53222            32532
                                          33225
                                          35223
                                          35322
                                          52233
                                          52332
                                          53223
                                          53232

Counting all permutations of prime factors gives A008480.
Compositions not of this type are counted by A349052/A129852/A129853.
Compositions of this type are counted by A349053, ranked by A349057.
The complement is counted by A349056.
Partitions of this type are counted by A349061, complement A349060.
The version counting patterns is A350138, complement A349058.
The version counting ordered factorizations is A350139, complement A349059.
The strong case is counted by A350251, complement A345164.
Positions of nonzero terms are A350353.
A001250 counts alternating permutations, complement A348615.
A025047 = alternating compositions, ranked by A345167, complement A345192.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A071321 gives the alternating sum of prime factors, reverse A071322.
A335452 counts anti-run permutations of prime factors, complement A336107.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A348379 counts factorizations with an alternating permutation.
Cf. A003242, A335433, A335448, A344614, A344652, A344653, A345173, A348613, A349798, A350252, A349800.

whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Permutations[Flatten[ConstantArray@@@ FactorInteger[n]]], !whkQ[#]&&!whkQ[-#]&]],{n,100}]

a(n) = A008480(n) - A349056(n).

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

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Dec 06 2021

nonn

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

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

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

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

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

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

cp's OEIS Frontend

A349053 Number of non-weakly alternating integer compositions of n.

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A349052 Number of weakly alternating compositions of n.

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A349057 Numbers k such that the k-th composition in standard order is not weakly alternating.

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

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

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A349058 Number of weakly alternating patterns of length n.

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

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A349797 Number of non-weakly alternating permutations of the multiset of prime factors of n.