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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.

%I A375219 #12 Aug 16 2024 14:46:26
%S A375219 19,1622,57,362997,6488,114,166336604,1814985,16220,190,136221590695,
%T A375219 998019624,5444955,32440,285,181552310074386,953551134865,3493068684,
%U A375219 12704895,56770,399,367942716863474473,1452418480595088,3814204539460,9314849824,25409790,90832,532
%N 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.
%C A375219 Trivially, T(n,n) = 1 and T(n,n-1) = 0.
%F A375219 Sum_{j=0..n-2} T(n,j) = (3*n)!/(6^n) - 1 = A014606(n) - 1.
%e A375219 The triangle begins
%e A375219          19;
%e A375219        1622,      57;
%e A375219      362997,    6488,   114,
%e A375219   166336604, 1814985, 16220, 190;
%e A375219 .
%e A375219 T(2,0) = 19: the permutations of {1,1,1,2,2,2} with no fixed triples are
%e A375219 [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].
%o A375219 (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]};
%o A375219 \\ returns row n of triangle, bsize is the block size in the multiset.
%o A375219 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}
%Y A375219 Cf. A014606.
%Y A375219 Cf. A374980, A375223 (columns 0 and 1 in a similar triangle for the multiset {1, 1, 2, 2, ..., n, n}).
%K A375219 nonn,tabl
%O A375219 2,1
%A A375219 _Hugo Pfoertner_, Aug 08 2024
%E A375219 More terms (three rows) from _Alois P. Heinz_, Aug 16 2024