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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
1, 2, 6, 18, 12, 24, 456, 5664, 20640, 576, 120, 13560, 1395840 (list; graph; refs; listen; history; text; internal format)



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.


Table of n, a(n) for n=1..13.


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


Triangle begins:



    6    18      12

   24   456    5664 20640 576

  120 13560 1395840 ?


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

Sequence in context: A340210 A165325 A316765 * A151870 A045962 A127257

Adjacent sequences:  A097632 A097633 A097634 * A097636 A097637 A097638




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


Entry revised Dec 31 2005



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Last modified September 23 05:33 EDT 2021. Contains 347609 sequences. (Running on oeis4.)