A325590 -id:A325590 - cp's OEIS frontend

A325589 Number of compositions of n whose circular differences are all 1 or -1.

0, 0, 2, 0, 2, 2, 2, 4, 4, 2, 8, 6, 8, 10, 12, 16, 18, 20, 28, 34, 42, 48, 62, 78, 92, 112, 146, 174, 216, 264, 326, 412, 500, 614, 770, 944, 1166, 1444, 1784, 2214, 2730, 3366, 4182, 5164, 6386, 7898, 9770, 12098, 14950, 18488, 22894, 28312, 35020, 43330, 53606

Offset: 1

Gus Wiseman, May 11 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

The circular differences of a composition c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

			The a(3) = 2 through a(11) = 8 compositions (empty columns not shown):
  (12)  (23)  (1212)  (34)  (1232)  (45)      (2323)  (56)
  (21)  (32)  (2121)  (43)  (2123)  (54)      (3232)  (65)
                            (2321)  (121212)          (121232)
                            (3212)  (212121)          (123212)
                                                      (212123)
                                                      (212321)
                                                      (232121)
                                                      (321212)

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000079, A008965, A034297, A173258, A325553, A325558, A325590, A325591.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ[1,##]&@@Abs[Differences[Append[#,First[#]]]]&]],{n,15}]

step(R,n,s)={matrix(n, n, i, j, if(i>j, if(j>s, R[i-j, j-s]) + if(j+s<=n, R[i-j, j+s])) )}
a(n)={sum(k=1, n, my(R=matrix(n,n,i,j,i==j&&abs(i-k)==1), t=0); while(R, R=step(R,n,1); t+=R[n,k]); t)} \\ Andrew Howroyd, Aug 23 2019

Terms a(26) and beyond from Andrew Howroyd, Aug 23 2019

A325556 Number of necklace compositions of n with distinct circular differences up to sign.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 7, 9, 13, 25, 27, 51, 63, 95, 123, 179, 205, 305, 409, 559, 715, 1009, 1337, 1869

Offset: 1

Gus Wiseman, May 11 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.

The circular differences of a composition c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

			The a(1) = 1 through a(10) = 13 necklace compositions:
  (1)  (2)  (3)  (4)  (5)  (6)  (7)    (8)     (9)     (A)
                                (124)  (125)   (126)   (127)
                                (142)  (134)   (162)   (136)
                                       (143)   (1125)  (145)
                                       (152)   (1134)  (154)
                                       (1124)  (1143)  (163)
                                       (1142)  (1152)  (172)
                                               (1224)  (235)
                                               (1422)  (253)
                                                       (1126)
                                                       (1162)
                                                       (1225)
                                                       (1522)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A008965, A235998, A318728, A325324, A325325, A325349, A325549, A325553, A325555, A325588, A325590.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Abs[Differences[Append[#,First[#]]]]&&neckQ[#]&]],{n,15}]

A325591 Number of compositions of n with circular differences all equal to 1, 0, or -1.

Original entry on oeis.org

1, 2, 4, 6, 11, 15, 27, 43, 68, 116, 189, 311, 519, 860, 1433, 2380, 3968, 6613, 11018, 18374, 30633, 51089, 85208, 142113, 237055, 395409, 659576, 1100262, 1835382, 3061711, 5107445, 8520122, 14213135, 23710173, 39553138, 65982316, 110071459, 183620990, 306316328

Offset: 1

Gus Wiseman, May 12 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.

The circular differences of a composition c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

			The a(1) = 1 through a(6) = 15 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (22)    (23)     (33)
             (21)   (112)   (32)     (222)
             (111)  (121)   (122)    (1122)
                    (211)   (212)    (1212)
                    (1111)  (221)    (1221)
                            (1112)   (2112)
                            (1121)   (2121)
                            (1211)   (2211)
                            (2111)   (11112)
                            (11111)  (11121)
                                     (11211)
                                     (12111)
                                     (21111)
                                     (111111)

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row sums of A309931.

Cf. A000079, A008965, A034297, A173258, A325351, A325551, A325553, A325558, A325589, A325590.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ[1,##]&@@Abs[DeleteCases[Differences[Append[#,First[#]]],0]]&]],{n,15}]

step(R,n,D)={matrix(n, n, i, j, if(i>j, sum(k=1, #D, my(s=D[k]); if(j>s && j+s<=n, R[i-j, j-s]))) )}
a(n)={sum(k=1, n, my(R=matrix(n,n,i,j,i==j&&abs(i-k)<=1), t=0); while(R, t+=R[n,k]; R=step(R,n,[0,1,-1])); t)} \\ Andrew Howroyd, Aug 23 2019

a(n) ~ c * d^n, where d = 1.66820206701846111636107... (see A034297), c = 0.65837031047271348106444... - Vaclav Kotesovec, Sep 21 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 23 2019

A325588 Number of necklace compositions of n with equal circular differences up to sign.

Original entry on oeis.org

1, 2, 3, 4, 4, 7, 5, 9, 8, 10, 8, 17, 9, 14, 15, 22, 12, 23, 14, 31, 23, 25, 19, 48, 25, 35, 36, 56, 33, 59, 43, 86, 64, 74, 76, 136, 95, 127, 138, 219, 178, 245, 249, 372, 370, 445, 506, 747, 730, 907, 1069, 1431, 1544, 1927, 2268, 2981, 3332, 4074, 4896, 6320

Offset: 1

Gus Wiseman, May 11 2019

nonn

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.

The circular differences of a sequence c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

			The a(1) = 1 through a(8) = 9 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (111)  (22)    (23)     (24)      (25)       (26)
                    (1111)  (11111)  (33)      (34)       (35)
                                     (222)     (1111111)  (44)
                                     (1212)               (1232)
                                     (111111)             (1313)
                                                          (2222)
                                                          (11111111)

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000079, A000740, A008965, A049988, A175342, A325549, A325556, A325558, A325590.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&SameQ@@Abs[Differences[Append[#,First[#]]]]&]],{n,15}]

step(R,n,s)={matrix(n, n, i, j, if(i>j, if(j>s, R[i-j, j-s]) + if(j+s<=n, R[i-j, j+s])) )}
w(n,s)={sum(k=1, n, my(R=matrix(n,n,i,j,i==j&&abs(i-k)==s), t=0, m=1); while(R, R=step(R,n,s); m++; t+=sumdiv(n, d, R[d,k]*d*eulerphi(n/d))/m ); t/n)}
a(n) = {numdiv(max(1,n)) + sum(s=1, n-1, w(n,s))} \\ Andrew Howroyd, Aug 24 2019

Terms a(26) and beyond from Andrew Howroyd, Aug 24 2019

A309937 Irregular triangle read by rows: T(n,k) is the number of compositions of n with 2k parts and circular differences all equal to 1 or -1, (n >= 3, 1 <= k <= n/3).

Original entry on oeis.org

2, 0, 2, 0, 2, 2, 0, 0, 4, 2, 0, 2, 0, 2, 0, 2, 0, 6, 0, 4, 0, 2, 2, 0, 6, 0, 0, 2, 0, 8, 2, 0, 8, 0, 2, 0, 4, 0, 12, 0, 2, 0, 6, 0, 10, 0, 2, 0, 16, 0, 2, 2, 0, 6, 0, 20, 0, 0, 4, 0, 18, 0, 12, 2, 0, 8, 0, 30, 0, 2, 0, 2, 0, 16, 0, 30, 0, 2, 0, 6, 0, 40, 0, 14, 0, 4, 0, 20, 0, 52, 0, 2

Offset: 3

Andrew Howroyd, Aug 23 2019

All values are even since the parts must alternate between even and odd and therefore a composition is never equal to its reversal.

The longest compositions will consist of alternating 1's and 2's. The number of parts cannot then exceed n / 3.

			Triangle begins:
  2;
  0;
  2;
  0, 2;
  2, 0;
  0, 4;
  2, 0, 2;
  0, 2, 0;
  2, 0, 6;
  0, 4, 0,  2;
  2, 0, 6,  0;
  0, 2, 0,  8;
  2, 0, 8,  0,  2;
  0, 4, 0, 12,  0;
  2, 0, 6,  0, 10;
  0, 2, 0, 16,  0,   2;
  2, 0, 6,  0, 20,   0;
  0, 4, 0, 18,  0,  12;
  2, 0, 8,  0, 30,   0,   2;
  0, 2, 0, 16,  0,  30,   0;
  2, 0, 6,  0, 40,   0,  14;
  0, 4, 0, 20,  0,  52,   0,   2;
  2, 0, 6,  0, 42,   0,  42,   0;
  0, 2, 0, 16,  0,  78,   0,  16;
  2, 0, 8,  0, 50,   0,  84,   0,  2;
  0, 4, 0, 18,  0,  96,   0,  56,  0;
  2, 0, 6,  0, 50,   0, 140,   0, 18;
  0, 2, 0, 16,  0, 116,   0, 128,  0, 2;
  ...
For n = 11 there are a total of 8 compositions:
  k = 1: (56), (65)
  k = 3: (121232), (123212), (212123), (212321), (232121), (321212)

Row sums are A325589.

Cf. A309931, A309938, A309939, A325558, A325590.

step(R,n)={matrix(n,n,i,j, if(i>j, if(j>1, R[i-j,j-1]) + if(j+1<=n, R[i-j,j+1])))}
T(n)={my(v=vector(n\3)); for(k=1, n, my(R=matrix(n,n,i,j,i==j&&abs(i-k)==1), m=0); while(R, m++; if(m%2==0, v[m/2]+=R[n,k]); R=step(R,n))); v}
for(n=3, 24, print(T(n)))

cp's OEIS Frontend

A325589 Number of compositions of n whose circular differences are all 1 or -1.

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A325556 Number of necklace compositions of n with distinct circular differences up to sign.

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A325591 Number of compositions of n with circular differences all equal to 1, 0, or -1.

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A325588 Number of necklace compositions of n with equal circular differences up to sign.

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Mathematica

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Extensions

A309937 Irregular triangle read by rows: T(n,k) is the number of compositions of n with 2k parts and circular differences all equal to 1 or -1, (n >= 3, 1 <= k <= n/3).

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