A117298 -id:A117298 - cp's OEIS frontend

A096441 Number of palindromic and unimodal compositions of n. Equivalently, the number of orbits under conjugation of even nilpotent n X n matrices.

1, 2, 2, 4, 3, 7, 5, 11, 8, 17, 12, 26, 18, 37, 27, 54, 38, 76, 54, 106, 76, 145, 104, 199, 142, 266, 192, 357, 256, 472, 340, 621, 448, 809, 585, 1053, 760, 1354, 982, 1740, 1260, 2218, 1610, 2818, 2048, 3559, 2590, 4485, 3264, 5616, 4097, 7018, 5120, 8728, 6378

Offset: 1

Nolan R. Wallach (nwallach(AT)ucsd.edu), Aug 10 2004

nonn

Number of partitions of n such that all differences between successive parts are even, see example. [Joerg Arndt, Dec 27 2012]
Number of partitions of n where either all parts are odd or all parts are even. - Omar E. Pol, Aug 16 2013
From Gus Wiseman, Jan 13 2022: (Start)
Also the number of integer partitions of n with all even multiplicities (or run-lengths) except possibly the first. These are the conjugates of the partitions described by Joerg Arndt above. For example, the a(1) = 1 through a(8) = 11 partitions are:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (11111)  (222)     (511)      (422)
                    (1111)           (411)     (31111)    (611)
                                     (2211)    (1111111)  (2222)
                                     (21111)              (3311)
                                     (111111)             (22211)
                                                          (41111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)
(End)

			From _Joerg Arndt_, Dec 27 2012: (Start)
There are a(10)=17 partitions of 10 where all differences between successive parts are even:
[ 1]  [ 1 1 1 1 1 1 1 1 1 1 ]
[ 2]  [ 2 2 2 2 2 ]
[ 3]  [ 3 1 1 1 1 1 1 1 ]
[ 4]  [ 3 3 1 1 1 1 ]
[ 5]  [ 3 3 3 1 ]
[ 6]  [ 4 2 2 2 ]
[ 7]  [ 4 4 2 ]
[ 8]  [ 5 1 1 1 1 1 ]
[ 9]  [ 5 3 1 1 ]
[10]  [ 5 5 ]
[11]  [ 6 2 2 ]
[12]  [ 6 4 ]
[13]  [ 7 1 1 1 ]
[14]  [ 7 3 ]
[15]  [ 8 2 ]
[16]  [ 9 1 ]
[17]  [ 10 ]
(End)

A. G. Elashvili and V. G. Kac, Classification of good gradings of simple Lie algebras. Lie groups and invariant theory, 85-104, Amer. Math. Soc. Transl. Ser. 2, 213, Amer. Math. Soc., Providence, RI, 2005.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Karin Baur and Nolan Wallach, Nice parabolic subalgebras of reductive Lie algebras, Represent. Theory 9 (2005), 1-29.
A. G. Elashvili and V. G. Kac, Classification of good gradings of simple Lie algebras, arXiv:math-ph/0312030, 2002-2004.
Sergi Elizalde and Emeric Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, U(0,z).

Bisections are A078408 and A096967.
The complement in partitions is counted by A006477
A version for compositions is A016116.
A pointed version is A035363, ranked by A066207.
A000041 counts integer partitions.
A025065 counts palindromic partitions.
A027187 counts partitions with even length/maximum.
A035377 counts partitions using multiples of 3.
A058696 counts partitions of even numbers, ranked by A300061.
A340785 counts factorizations into even factors.
Cf. A000009, A002865, A027383, A035457, A117298, A117989, A168021, A274230, A345170, A349060, A349061.

b:= proc(n, i) option remember; `if`(i>n, 0,
      `if`(irem(n, i)=0, 1, 0) +add(`if`(irem(j, 2)=0,
       b(n-i*j, i+1), 0), j=0..n/i))
    end:
a:= n-> b(n, 1):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 26 2014

(* The following Mathematica program first generates all of the palindromic, unimodal compositions of n and then counts them. *)
Pal[n_] := Block[{i, j, k, m, Q, L}, If[n == 1, Return[{{1}}]]; If[n == 2, Return[{{1, 1}, {2}}]]; L = {{n}}; If[Mod[n, 2] == 0, L = Append[L, {n/2, n/2}]]; For[i = 1, i < n, i++, Q = Pal[n - 2i]; m = Length[Q]; For[j = 1, j <= m, j++, If[i <= Q[[j, 1]], L = Append[L, Append[Prepend[Q[[j]], i], i]]]]]; L] NoPal[n_] := Length[Pal[n]]
a[n_] := PartitionsQ[n] + If[EvenQ[n], PartitionsP[n/2], 0]; Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Mar 17 2014, after Vladeta Jovovic *)
Table[Length[Select[IntegerPartitions[n],And@@EvenQ/@Rest[Length/@Split[#]]&]],{n,1,30}] (* Gus Wiseman, Jan 13 2022 *)

my(x='x+O('x^66)); Vec(eta(x^2)/eta(x)+1/eta(x^2)-2) \\ Joerg Arndt, Jan 17 2016

G.f.: sum(j>=1, q^j * (1-q^j)/prod(i=1..j, 1-q^(2*i) ) ).
G.f.: F + G - 2, where F = Product_{j>=1} 1/(1-q^(2*j)), G = Product_{j>=0} 1/(1-q^(2*j+1)).
a(2*n) = A000041(n) + A000009(2*n); a(2*n-1) = A000009(2*n-1). - Vladeta Jovovic, Aug 11 2004
a(n) = A000009(n) + A035363(n) = A000041(n) - A006477(n). - Omar E. Pol, Aug 16 2013

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}]

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).

A349796 Number of non-strict 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, 0, 1, 2, 5, 8, 15, 23, 37, 52, 80, 109, 156, 208, 289, 378, 509, 654, 865, 1098, 1425, 1789, 2290, 2852, 3603, 4450, 5569, 6830, 8467, 10321, 12701, 15393, 18805, 22678, 27535, 33057, 39908, 47701, 57304, 68226, 81572, 96766, 115212, 136201

Offset: 0

Gus Wiseman, Dec 25 2021

nonn

Also the number of non-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 involves the somewhat degenerate case where no strict increases are allowed.

			The a(7) = 1 through a(11) = 15 partitions:
  (3211)  (4211)   (3321)    (5311)     (4322)
          (32111)  (4311)    (6211)     (4421)
                   (5211)    (32221)    (5411)
                   (42111)   (33211)    (6311)
                   (321111)  (43111)    (7211)
                             (52111)    (42221)
                             (421111)   (43211)
                             (3211111)  (53111)
                                        (62111)
                                        (322211)
                                        (332111)
                                        (431111)
                                        (521111)
                                        (4211111)
                                        (32111111)

Counting all non-strict partitions gives A047967.
Signatures of this type are counted by A274230, complement A027383.
The strict instead of non-strict version is A347548, ranked by A350352.
The version for compositions allowing strict is A349053, ranked by A349057.
Allowing strict partitions gives A349061, complement A349060.
The complement in non-strict partitions is A349795.
These partitions are ranked by A350140 = A349794 \ A005117.
A000041 = integer partitions, strict A000009.
A001250 = alternating permutations, complement A348615.
A003242 = Carlitz (anti-run) compositions.
A025047 = alternating compositions, ranked by A345167.
A025048/A025049 = directed alternating compositions.
A096441 = weakly alternating 0-appended partitions.
A345170 = partitions w/ an alternating permutation, ranked by A345172.
A349052 = weakly alternating compositions.
A349056 = weakly alternating permutations of prime indices.
A349798 = weakly but not strongly alternating permutations of prime indices.
Cf. A000111, A002865, A117298, A117989, A129852, A129853, A345165, A345192, A349054, A349059, A349801.

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[IntegerPartitions[n],!whkQ[#]&&!whkQ[-#]&&!UnsameQ@@#&]],{n,0,30}]

a(n) = A349061(n) - A347548(n).

A349794 Numbers whose prime signature has an odd term other than the first or last.

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 102, 105, 110, 114, 120, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 182, 186, 190, 195, 204, 210, 220, 222, 228, 230, 231, 238, 240, 246, 255, 258, 260, 264, 266, 270, 273, 276, 280, 282, 285, 286, 290, 294, 300, 308

Offset: 1

Gus Wiseman, Dec 06 2021

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Also numbers whose multiset of prime factors is not weakly alternating, 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 decreases are allowed.

			The terms and their prime indices begin:
   30: {1,2,3}
   42: {1,2,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
  102: {1,2,7}
  105: {2,3,4}
  110: {1,3,5}
  114: {1,2,8}
  120: {1,1,1,2,3}
  130: {1,3,6}
  132: {1,1,2,5}
  138: {1,2,9}

The complement for compositions is A025047, ranked by A345167.
Signatures of this type are counted by A274230, complement A027383.
The strong case is A289553, complement A167171.
The strong case for compositions is A345192, ranked by A345168.
The version for compositions is A349053, ranked by A349057.
These partitions are counted by A349061, complement A349060, strong A349801.
The non-strict case is counted by A349795.
A001250 counts alternating permutations, complement A348615.
A096441 counts weakly alternating partitions if 0 is appended.
A345164 counts alternating permutations of prime indices, weak A349056.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349052 counts weakly alternating compositions.
A349059 counts weakly alternating ordered factorizations, strong A348610.
Cf. A003242, A112798, A114901, A117298, A128761, A129852, A129853, A344614, A344615, A345165, A349796, A349798.

Select[Range[100],PrimeNu[#]>1&&!And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}]&]

A349801 Number of integer partitions of n into three or more parts or into two equal parts.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 11, 18, 25, 37, 50, 71, 94, 128, 168, 223, 288, 376, 480, 617, 781, 991, 1243, 1563, 1945, 2423, 2996, 3704, 4550, 5589, 6826, 8333, 10126, 12293, 14865, 17959, 21618, 25996, 31165, 37318, 44562, 53153, 63239, 75153, 89111, 105535, 124730

Offset: 0

Gus Wiseman, Dec 23 2021

nonn

This sequence arose as the following degenerate case. If we define a sequence to be alternating if it is alternately strictly increasing and strictly decreasing, starting with either, then a(n) is the number of non-alternating integer partitions of n. Under this interpretation:
- The non-strict case is A047967, weak A349796, weak complement A349795.
- The complement is counted by A065033(n) = ceiling(n/2) for n > 0.
- These partitions are ranked by A289553 \ {1}, complement A167171 \/ {1}.
- The version for compositions is A345192, ranked by A345168.
- The weak version for compositions is A349053, ranked by A349057.
- The weak version is A349061, complement A349060, ranked by A349794.

			The a(2) = 1 through a(7) = 11 partitions:
  (11)  (111)  (22)    (221)    (33)      (322)
               (211)   (311)    (222)     (331)
               (1111)  (2111)   (321)     (421)
                       (11111)  (411)     (511)
                                (2211)    (2221)
                                (3111)    (3211)
                                (21111)   (4111)
                                (111111)  (22111)
                                          (31111)
                                          (211111)
                                          (1111111)

A000041 counts partitions, ordered A011782.
A001250 counts alternating permutations, complement A348615.
A004250 counts partitions into three or more parts, strict A347548.
A025047/A025048/A025049 count alternating compositions, ranked by A345167.
A096441 counts weakly alternating 0-appended partitions.
A345165 counts partitions w/ no alternating permutation, complement A345170.
Cf. A000070, A001700, A002865, A117298, A117989, A102726, A128761, A345162, A345163, A345166, A349798.

Table[Length[Select[IntegerPartitions[n],MatchQ[#,{x_,x_}|{,,__}]&]],{n,0,10}]

a(1) = 0; a(n > 0) = A000041(n) - ceiling(n/2).

A350137 Nonsquarefree numbers whose prime signature, except possibly the first and last parts, is all even.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 63, 64, 68, 72, 75, 76, 80, 81, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 121, 124, 125, 126, 128, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 169, 171, 172

Offset: 1

Gus Wiseman, Dec 23 2021

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
Also nonsquarefree numbers whose prime factors, taken in order and with multiplicity, are alternately constant and weakly increasing, starting with either.
Also the Heinz numbers of non-strict integer partitions whose part multiplicities, except possibly the first and last, are all even. These are counted by A349795.

			The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}

This is the nonsquarefree case of the complement of A349794.
These are the Heinz numbers of the partitions counted by A349795.
A version for compositions is A349799, counted by A349800.
A complementary version is A350140, counted by A349796.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A005117 = squarefree numbers, complement A013929.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A124010 = prime signature, sorted A118914.
A345164 = alternating permutations of prime indices, complement A350251.
A349052/A129852/A129853 = weakly alternating compositions.
A349053 = non-weakly alternating compositions, ranked by A349057.
A349056 = weakly alternating permutations of prime indices.
A349058 = weakly alternating patterns, complement A350138.
A349060 = weakly alternating partitions, complement A349061.
Cf. A000111, A096441, A117298, A335433, A335448, A335452, A344614, A344652, A344653, A345173, A349059, A349797.

Select[Range[100],!SquareFreeQ[#]&&(PrimePowerQ[#]||And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}])&]

cp's OEIS Frontend

A096441 Number of palindromic and unimodal compositions of n. Equivalently, the number of orbits under conjugation of even nilpotent n X n matrices.

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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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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.

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A349796 Number of non-strict integer partitions of n with at least one part of odd multiplicity that is not the first or last.

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A349794 Numbers whose prime signature has an odd term other than the first or last.

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A349801 Number of integer partitions of n into three or more parts or into two equal parts.

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A350137 Nonsquarefree numbers whose prime signature, except possibly the first and last parts, is all even.

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