A375219 T(n,k) is the number of permutations of the multiset {1, 1, 1, 2, 2, 2, ..., n, n, n} with k occurrences of fixed triples (j,j,j), where T(n,k), n >= 2, 0 <= k <= n-2 is a triangle read by rows.

19, 1622, 57, 362997, 6488, 114, 166336604, 1814985, 16220, 190, 136221590695, 998019624, 5444955, 32440, 285, 181552310074386, 953551134865, 3493068684, 12704895, 56770, 399, 367942716863474473, 1452418480595088, 3814204539460, 9314849824, 25409790, 90832, 532

Offset

2,1

Comments

Trivially, T(n,n) = 1 and T(n,n-1) = 0.

Links

Table of n, a(n) for n=2..29.

Formula

Sum_{j=0..n-2} T(n,j) = (3*n)!/(6^n) - 1 = A014606(n) - 1.

Example

The triangle begins
         19;
       1622,      57;
     362997,    6488,   114,
  166336604, 1814985, 16220, 190;
.
T(2,0) = 19: the permutations of {1,1,1,2,2,2} with no fixed triples are
[1,1,2,1,2,2], [1,1,2,2,1,2], [1,1,2,2,2,1], [1,2,1,1,2,2], [1,2,1,2,1,2], [1,2,1,2,2,1], [1,2,2,1,1,2], [1,2,2,1,2,1], [1,2,2,2,1,1], [2,1,1,1,2,2], [2,1,1,2,1,2], [2,1,1,2,2,1], [2,1,2,1,1,2], [2,1,2,1,2,1], [2,1,2,2,1,1], [2,2,1,1,1,2], [2,2,1,1,2,1], [2,2,1,2,1,1], [2,2,2,1,1,1].

Prog

(PARI) mima (x, n1=1, i2=-oo) = {my (n2, n=#x, mi=x[n1], ma=mi); n2=if (i2<=0, n, min(n, i2)); for (i=n1+1, n2, if (x[i]<mi, mi=x[i], if (x[i]>ma, ma=x[i]))); [mi, ma]};
\\ returns row n of triangle, bsize is the block size in the multiset.
a375219(n, bsize=3) = {my (p=vector(bsize*n, i, 1+(i-1)\bsize), r=s=vector(n), m=vector(n-1)); forperm (p, q, for (b=1, n, my (bm=bsize*(b-1), j=mima(q, bm+1, bm+bsize)); r[b]=j[1]; s[b]=j[2]); my (rs=vector(n, i, r[i]==i && s[i]==i)); for (k=0 , n-2, m[k+1]+=vecsum(rs)==k)); m}

Crossrefs

Cf. A014606.

Cf. A374980, A375223 (columns 0 and 1 in a similar triangle for the multiset {1, 1, 2, 2, ..., n, n}).

Sequence in context: A223498 A352465 A054949 * A242564 A068748 A289736

Adjacent sequences: A375216 A375217 A375218 * A375220 A375221 A375222

Keyword

nonn,tabl

Author

Hugo Pfoertner, Aug 08 2024

Extensions

More terms (three rows) from Alois P. Heinz, Aug 16 2024

Status

approved