A325789 -id:A325789 - cp's OEIS frontend

A325791 Number of necklace permutations of {1..n} such that every positive integer from 1 to n * (n + 1)/2 is the sum of some circular subsequence.

1, 1, 1, 2, 4, 20, 82, 252, 1074, 7912, 39552, 152680, 776094, 5550310, 30026848, 108376910

Offset: 0

Gus Wiseman, May 23 2019

A necklace permutation is a permutation that is either empty or whose first part is the minimum. A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. The only circular subsequence of maximum length is the sequence itself, not any rotation of it. For example, the circular subsequences of (1,3,2) are: (), (1), (2), (3), (1,3), (2,1), (3,2), (1,3,2).

			The a(1) = 1 through a(5) = 20 permutations:
  (1)  (1,2)  (1,2,3)  (1,2,3,4)  (1,2,3,4,5)
              (1,3,2)  (1,3,2,4)  (1,2,3,5,4)
                       (1,4,2,3)  (1,2,4,3,5)
                       (1,4,3,2)  (1,2,4,5,3)
                                  (1,2,5,4,3)
                                  (1,3,2,5,4)
                                  (1,3,4,2,5)
                                  (1,3,4,5,2)
                                  (1,3,5,2,4)
                                  (1,3,5,4,2)
                                  (1,4,2,3,5)
                                  (1,4,2,5,3)
                                  (1,4,3,2,5)
                                  (1,4,5,2,3)
                                  (1,4,5,3,2)
                                  (1,5,2,3,4)
                                  (1,5,2,4,3)
                                  (1,5,3,2,4)
                                  (1,5,3,4,2)
                                  (1,5,4,3,2)

Cf. A002033, A008965, A032153, A103295, A126796, A304793, A325684, A325781, A325786, A325788, A325789, A325790.

subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Permutations[Range[n]],#=={}||First[#]==1&&Range[n*(n+1)/2]==Union[Total/@subalt[#]]&]],{n,0,5}]

a(11)-a(15) from Bert Dobbelaere, Nov 01 2020

A325786 Number of complete necklace compositions of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 7, 12, 19, 41, 71, 141, 255, 509, 924, 1882, 3395, 6838, 12715, 25233, 47049

Offset: 1

Gus Wiseman, May 22 2019

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. A circular subsequence is a sequence of consecutive terms where the first and last parts are also considered consecutive. A necklace composition of n is complete if every positive integer from 1 to n is the sum of some circular subsequence.

			The a(1) = 1 through a(8) = 19 necklace compositions:
  (1)  (11)  (12)   (112)   (113)    (123)     (124)      (1124)
             (111)  (1111)  (122)    (132)     (142)      (1133)
                            (1112)   (1113)    (1114)     (1142)
                            (11111)  (1122)    (1123)     (1214)
                                     (1212)    (1132)     (1223)
                                     (11112)   (1213)     (1322)
                                     (111111)  (1222)     (11114)
                                               (11113)    (11123)
                                               (11122)    (11132)
                                               (11212)    (11213)
                                               (111112)   (11222)
                                               (1111111)  (11312)
                                                          (12122)
                                                          (111113)
                                                          (111122)
                                                          (111212)
                                                          (112112)
                                                          (1111112)
                                                          (11111111)

Cf. A000740, A002033, A008965, A103295, A108917, A126796, A276024, A325549, A325682, A325781, A325788, A325789, A325791.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&Union[Total/@subalt[#]]==Range[n]&]],{n,15}]

A325787 Number of perfect strict necklace compositions of n.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 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, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, May 22 2019

nonn

A strict necklace composition of n is a finite sequence of distinct positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. In other words, it is a strict composition of n starting with its least part. A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. A necklace composition of n is perfect if every positive integer from 1 to n is the sum of exactly one distinct circular subsequence. For example, the composition (1,2,6,4) is perfect because it has the following circular subsequences and sums:
(1)
(2)
(1,2)
(4)
(4,1)
(6)
(4,1,2)
(2,6)
(1,2,6)
(6,4)
(6,4,1)
(2,6,4)
(1,2,6,4)
a(n) > 0 iff n = A002061(k) = A004136(k) for some k. - Bert Dobbelaere, Nov 11 2020

			The a(1) = 1 through a(31) = 10 perfect strict necklace compositions (empty columns not shown):
  (1)  (1,2)  (1,2,4)  (1,2,6,4)  (1,3,10,2,5)  (1,10,8,7,2,3)
              (1,4,2)  (1,3,2,7)  (1,5,2,10,3)  (1,13,6,4,5,2)
                       (1,4,6,2)                (1,14,4,2,3,7)
                       (1,7,2,3)                (1,14,5,2,6,3)
                                                (1,2,5,4,6,13)
                                                (1,2,7,4,12,5)
                                                (1,3,2,7,8,10)
                                                (1,3,6,2,5,14)
                                                (1,5,12,4,7,2)
                                                (1,7,3,2,4,14)
From _Bert Dobbelaere_, Nov 11 2020: (Start)
Compositions matching nonzero terms from a(57) to a(273), up to symmetry.
a(57) = 12:
  (1,2,10,19,4,7,9,5)
  (1,3,5,11,2,12,17,6)
  (1,3,8,2,16,7,15,5)
  (1,4,2,10,18,3,11,8)
  (1,4,22,7,3,6,2,12)
  (1,6,12,4,21,3,2,8)
a(73) = 8:
  (1,2,4,8,16,5,18,9,10)
  (1,4,7,6,3,28,2,8,14)
  (1,6,4,24,13,3,2,12,8)
  (1,11,8,6,4,3,2,22,16)
a(91) = 12:
  (1,2,6,18,22,7,5,16,4,10)
  (1,3,9,11,6,8,2,5,28,18)
  (1,4,2,20,8,9,23,10,3,11)
  (1,4,3,10,2,9,14,16,6,26)
  (1,5,4,13,3,8,7,12,2,36)
  (1,6,9,11,29,4,8,2,3,18)
a(133) = 36:
  (1,2,9,8,14,4,43,7,6,10,5,24)
  (1,2,12,31,25,4,9,10,7,11,16,5)
  (1,2,14,4,37,7,8,27,5,6,13,9)
  (1,2,14,12,32,19,6,5,4,18,13,7)
  (1,3,8,9,5,19,23,16,13,2,28,6)
  (1,3,12,34,21,2,8,9,5,6,7,25)
  (1,3,23,24,6,22,10,11,18,2,5,8)
  (1,4,7,3,16,2,6,17,20,9,13,35)
  (1,4,16,3,15,10,12,14,17,33,2,6)
  (1,4,19,20,27,3,6,25,7,8,2,11)
  (1,4,20,3,40,10,9,2,15,16,6,7)
  (1,5,12,21,29,11,3,16,4,22,2,7)
  (1,7,13,12,3,11,5,18,4,2,48,9)
  (1,8,10,5,7,21,4,2,11,3,26,35)
  (1,14,3,2,4,7,21,8,25,10,12,26)
  (1,14,10,20,7,6,3,2,17,4,8,41)
  (1,15,5,3,25,2,7,4,6,12,14,39)
  (1,22,14,20,5,13,8,3,4,2,10,31)
a(183) = 40:
  (1,2,13,7,5,14,34,6,4,33,18,17,21,8)
  (1,2,21,17,11,5,9,4,26,6,47,15,12,7)
  (1,2,28,14,5,6,9,12,48,18,4,13,16,7)
  (1,3,5,6,25,32,23,10,18,2,17,7,22,12)
  (1,3,12,7,20,14,44,6,5,24,2,28,8,9)
  (1,3,24,6,12,14,11,55,7,2,8,5,16,19)
  (1,4,6,31,3,13,2,7,14,12,17,46,8,19)
  (1,4,8,52,3,25,18,2,9,24,6,10,7,14)
  (1,4,20,2,12,3,6,7,33,11,8,10,35,31)
  (1,5,2,24,15,29,14,21,13,4,33,3,9,10)
  (1,5,23,27,42,3,4,11,2,19,12,10,16,8)
  (1,6,8,22,4,5,33,21,3,20,32,16,2,10)
  (1,8,3,10,23,5,56,4,2,14,15,17,7,18)
  (1,8,21,45,6,7,11,17,3,2,10,4,23,25)
  (1,9,5,40,3,4,21,35,16,18,2,6,11,12)
  (1,9,14,26,4,2,11,5,3,12,27,34,7,28)
  (1,9,21,25,3,4,8,5,6,16,2,36,14,33)
  (1,10,22,34,27,12,3,4,2,14,24,5,8,17)
  (1,10,48,9,19,4,8,6,7,17,3,2,34,15)
  (1,12,48,6,2,38,3,22,7,10,11,5,4,14)
a(273) = 12:
  (1,2,4,8,16,32,27,26,11,9,45,13,10,29,5,17,18)
  (1,3,12,10,31,7,27,2,6,5,19,20,62,14,9,28,17)
  (1,7,3,15,33,5,24,68,2,14,6,17,4,9,19,12,34)
  (1,7,12,44,25,41,9,17,4,6,22,33,13,2,3,11,23)
  (1,7,31,2,11,3,9,36,17,4,22,6,18,72,5,10,19)
  (1,21,11,50,39,13,6,4,14,16,25,26,3,2,7,8,27)
(End)

Bert Dobbelaere, Table of n, a(n) for n = 1..306

Cf. A002033, A002061, A004136, A008965, A032153, A103300, A126796, A325680, A325682, A325780, A325782, A325786, A325788, A325789.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],neckQ[#]&&Sort[Total/@subalt[#]]==Range[n]&]],{n,30}]

More terms from Bert Dobbelaere, Nov 11 2020

A325990 Numbers with more than one perfect factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 256, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456

Offset: 1

Gus Wiseman, May 30 2019

nonn

First differs from A060476 in lacking 1 and having 432.

A perfect factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of exactly one submultiset of the factors. This is the intersection of covering (or complete) factorizations (A325988) and knapsack factorizations (A292886).

Positions of terms > 1 in A325989.

Cf. A002033, A292886, A325780, A325781, A325782, A325787, A325789, A325988.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Function[n,Length[Select[facs[n],Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]]>1]]

A325989 Number of perfect factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, May 30 2019

nonn

			The a(216) = 4 perfect factorizations:
  (2*2*2*3*3*3)
  (2*2*2*3*9)
  (2*3*3*3*4)
  (2*3*4*9)

Positions of terms > 1 are A325990.

Cf. A002033, A103300, A292886, A325685, A325780, A325787, A325789, A325988.

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],Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]],{n,100}]

a(2^n) = A002033(n).

cp's OEIS Frontend

A325791 Number of necklace permutations of {1..n} such that every positive integer from 1 to n * (n + 1)/2 is the sum of some circular subsequence.

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A325786 Number of complete necklace compositions of n.

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A325787 Number of perfect strict necklace compositions of n.

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A325990 Numbers with more than one perfect factorization.

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A325989 Number of perfect factorizations of n.

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