cp's OEIS Frontend

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.

%I A334778 #16 May 17 2020 13:50:15
%S A334778 1,0,1,0,4,2,0,18,66,6,0,72,1168,1192,88,0,270,16220,61830,33600,1480,
%T A334778 0,972,202416,2150688,3821760,1268292,40272,0,3402,2395540,62178928,
%U A334778 272509552,279561086,62954948,1476944,0,11664,27517568,1629254640,15313310208,36381368048,24342647424,3963672720,71865728
%N 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.
%C A334778 T(n,k) is divisible by n for n > 0.
%H A334778 Andrew Howroyd, <a href="/A334778/b334778.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F A334778 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.
%F A334778 A334780(n) = Sum_{k=1..n} k*T(n,k).
%e A334778 Triangle begins:
%e A334778    1;
%e A334778    0,    1;
%e A334778    0,    4,       2;
%e A334778    0,   18,      66,        6;
%e A334778    0,   72,    1168,     1192,        88;
%e A334778    0,  270,   16220,    61830,     33600,      1480;
%e A334778    0,  972,  202416,  2150688,   3821760,   1268292,    40272;
%e A334778    0, 3402, 2395540, 62178928, 272509552, 279561086, 62954948, 1476944;
%e A334778   ...
%e A334778 The T(2,1) = 4 permutations of 1122 with 1 local maximum are 1122, 1221, 2112, 2211.
%e A334778 The T(2,2) = 2 permutations of 1122 with 2 local maxima are 1212, 2121.
%o A334778 (PARI)
%o A334778 CircPeaksBySig(sig, D)={
%o A334778   my(F(lev,p,q) = my(key=[lev,p,q], z); if(!mapisdefined(FC, key, &z),
%o A334778     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) )));
%o A334778     mapput(FC, key, z)); z);
%o A334778   local(FC=Map());
%o A334778   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)));
%o A334778 }
%o A334778 Row(n)={ if(n==0, [1], CircPeaksBySig(vector(n,i,2), [0..n])) }
%o A334778 { for(n=0, 8, print(Row(n))) }
%Y A334778 Columns k=0..6 are A000007, A027261(n-1), A159716, A159717, A159718, A159719, A159720.
%Y A334778 Row sums are A000680.
%Y A334778 Main diagonal is A334779.
%Y A334778 The version for permutations of 1..n is A263789.
%Y A334778 Cf. A334218, A334774, A334780.
%K A334778 nonn,tabl
%O A334778 0,5
%A A334778 _Andrew Howroyd_, May 13 2020