A097635 - OEIS

%I #9 Mar 28 2020 04:31:07

%S 1,2,6,18,12,24,456,5664,20640,576,120,13560,1395840

%N Triangle read by rows: T(n,k) = number of unique-valued sequences of length k, n >= 1, 1 <= k <= 2n-3, in the symmetric group S_n.

%C Definition: Let G(*) be a semigroup. A finite sequence u_1, u_2,... u_n of elements from G is called unique-valued with value v and length n provided v = u_1 * u_2 *... * u_n and v != u_p(1) * u_p(2) *... * u_p(n) for any non-identity permutation p of the indices {1, 2,... n}.

%C In other words, the only way to obtain the unique value v from the elements u_1, u_2,... u_n is by multiplying them in that particular order; any other order always gives a value different from v.

%C When the length of the sequence is 2, the meaning of "unique-valued" is equivalent to "the two elements do not commute under *".

%C I proved that the maximal possible length of a unique-valued sequence in the monoid M_n(*) of all n X n matrices (with entries in some ring with 1) is exactly 2n-1 (that 2n-1 is an upper bound follows from the Amitsur-Levitzki theorem), providing a positive example that this limit is reached.

%C I also proved that the maximal possible length of unique-valued sequences in S_n is 2n-3 (using n-simplex and again using the Amitsur-Levitzki theorem), but didn't find examples that this limit is really reached. My computer said "yes" for n=2 to 5, but even 6 is too large to compute.

%C The numbers of unique-valued sequences in S_n of the maximal length 2n-3 form a sequence 1, 2, 12, 576, which seems to coincide with A002860, the number of Latin squares of order n.

%F a(n*(n-1)/2) = n!.

%e Triangle begins:

%e     1

%e     2

%e     6    18      12

%e    24   456    5664 20640 576

%e   120 13560 1395840 ?

%Y Possibly related to A002860 (the number of Latin Squares) or A052129.

%K nonn,nice,more,tabf

%O 1,2

%A Aleksandar Blazhevski-Cane (CaneB(AT)mt.net.mk), Aug 17 2004

%E Entry revised Dec 31 2005