A097635 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.

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

Offset: 1

Author

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

Keywords

• nonn
• nice
• more
• tabf

Comments

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}.

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.

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

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.

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.

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.

Examples

			Triangle begins:
    1
    2
    6    18      12
   24   456    5664 20640 576
  120 13560 1395840 ?
		

Crossrefs

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

Formula

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

Extensions

Entry revised Dec 31 2005