A325558 - OEIS

A325558

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

%I #14 Aug 24 2019 11:51:15

%S 1,2,4,5,6,10,8,16,13,16,18,32,20,30,30,57,34,52,46,96,74,86,84,174,

%T 119,170,192,306,244,332,372,628,560,694,812,1259,1228,1566,1852,2696,

%U 2806,3538,4260,5894,6482,8098,9890,13392,15049,18706,23018,30298,35198

%N Number of compositions of n with equal circular differences up to sign.

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

%C 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).

%H Andrew Howroyd, <a href="/A325558/b325558.txt">Table of n, a(n) for n = 1..200</a>

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e The a(1) = 1 through a(8) = 16 compositions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (12) (13) (14) (15) (16) (17)

%e (21) (22) (23) (24) (25) (26)

%e (111) (31) (32) (33) (34) (35)

%e (1111) (41) (42) (43) (44)

%e (11111) (51) (52) (53)

%e (222) (61) (62)

%e (1212) (1111111) (71)

%e (2121) (1232)

%e (111111) (1313)

%e (2123)

%e (2222)

%e (2321)

%e (3131)

%e (3212)

%e (11111111)

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

%o (PARI)

%o 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])) )}

%o w(n,k,s)={my(R=matrix(n,n,i,j,i==j&&abs(i-k)==s), t=0); while(R, R=step(R,n,s); t+=R[n,k]); t}

%o a(n) = {numdiv(max(1,n)) + sum(s=1, n-1, sum(k=1, n, w(n,k,s)))} \\ _Andrew Howroyd_, Aug 22 2019

%Y Cf. A000079, A008965, A047966, A049988, A098504, A173258, A175342, A325553, A325557, A325588, A325589.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 11 2019

%E a(26)-a(42) from _Lars Blomberg_, May 30 2019

%E Terms a(43) and beyond from _Andrew Howroyd_, Aug 22 2019