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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. 19
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 (list; table; graph; refs; listen; history; text; internal format)
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.
Sequence in context: A244131 A206428 A357012 * A111549 A279411 A022696
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, May 13 2020
STATUS
approved

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Last modified June 20 07:26 EDT 2024. Contains 373512 sequences. (Running on oeis4.)