A334778 Triangle read by rows: T(n,k) is the number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly k local maxima.

1, 0, 1, 0, 4, 2, 0, 18, 66, 6, 0, 72, 1168, 1192, 88, 0, 270, 16220, 61830, 33600, 1480, 0, 972, 202416, 2150688, 3821760, 1268292, 40272, 0, 3402, 2395540, 62178928, 272509552, 279561086, 62954948, 1476944, 0, 11664, 27517568, 1629254640, 15313310208, 36381368048, 24342647424, 3963672720, 71865728

Offset

0,5

Comments

T(n,k) is divisible by n for n > 0.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Formula

T(n,k) = n*(2*F(2,n-1,k-1,0) + F(2,n-1,k-2,1)) for n > 1 where F(m,n,p,q) = Sum_{i=0..p} Sum_{j=0..min(m-i, q)} F(m, n-1, p-i, q-j+i) * binomial(m+2*(q-j)+1, 2*q+i-j+1) * binomial(q-j+i, i) * binomial(q+1, j) for n > 1 with F(m,1,0,q) = binomial(m-1, q), F(m,1,p,q) = 0 for p > 0.

A334780(n) = Sum_{k=1..n} k*T(n,k).

Example

Triangle begins:
   1;
   0,    1;
   0,    4,       2;
   0,   18,      66,        6;
   0,   72,    1168,     1192,        88;
   0,  270,   16220,    61830,     33600,      1480;
   0,  972,  202416,  2150688,   3821760,   1268292,    40272;
   0, 3402, 2395540, 62178928, 272509552, 279561086, 62954948, 1476944;
  ...
The T(2,1) = 4 permutations of 1122 with 1 local maximum are 1122, 1221, 2112, 2211.
The T(2,2) = 2 permutations of 1122 with 2 local maxima are 1212, 2121.

Prog

(PARI)
CircPeaksBySig(sig, D)={
  my(F(lev, p, q) = my(key=[lev, p, q], z); if(!mapisdefined(FC, key, &z),
    my(m=sig[lev]); z = if(lev==1, if(p==0, binomial(m-1, q), 0), sum(i=0, p, sum(j=0, min(m-i, q), self()(lev-1, p-i, q-j+i) * binomial(m+2*(q-j)+1, 2*q+i-j+1) * binomial(q-j+i, i) * binomial(q+1, j) )));
    mapput(FC, key, z)); z);
  local(FC=Map());
  vector(#D, i, my(k=D[i], lev=#sig); if(lev==1, k==1, my(m=sig[lev]); lev*sum(j=1, min(m, k), m*binomial(m-1, j-1)*F(lev-1, k-j, j-1)/j)));
}
Row(n)={ if(n==0, [1], CircPeaksBySig(vector(n, i, 2), [0..n])) }
{ for(n=0, 8, print(Row(n))) }

Crossrefs

Columns k=0..6 are A000007, A027261(n-1), A159716, A159717, A159718, A159719, A159720.

Row sums are A000680.

Main diagonal is A334779.

The version for permutations of 1..n is A263789.

Cf. A334218, A334774, A334780.

Sequence in context: A244131 A206428 A357012 * A111549 A279411 A022696

Adjacent sequences: A334775 A334776 A334777 * A334779 A334780 A334781

Keyword

nonn,tabl

Author

Andrew Howroyd, May 13 2020

Status

approved