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subsequence sum
Davenport's constant ★★★
Author(s):
For a finite (additive) abelian group , the Davenport constant of
, denoted
, is the smallest integer
so that every sequence of elements of
with length
has a nontrivial subsequence which sums to zero.
Conjecture
![$ s( {\mathbb Z}_n^d) = d(n-1) + 1 $](/files/tex/2bea401874dc0a9cd37e10b6df927b12a7ce2402.png)
Keywords: Davenport constant; subsequence sum; zero sum
Gao's theorem for nonabelian groups ★★
Author(s): DeVos
For every finite multiplicative group , let
(
) denote the smallest integer
so that every sequence of
elements of
has a subsequence of length
(length
) which has product equal to 1 in some order.
Conjecture
for every finite group
.
![$ s'(G) = s(G) + |G| - 1 $](/files/tex/91264d9b321ddb6a08656e574ccf314d9ebfa121.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
Keywords: subsequence sum; zero sum
Few subsequence sums in Z_n x Z_n ★★
Conjecture For every
, the sequence in
consisting of
copes of
and
copies of
has the fewest number of distinct subsequence sums over all zero-free sequences from
of length
.
![$ 0 \le t \le n-1 $](/files/tex/3f439dde8bba9a34c4b73a7bf35d2ba2d600dd53.png)
![$ {\mathbb Z}_n^2 $](/files/tex/784102d74f41429c112d0dd6746a4ab9f1957afe.png)
![$ n-1 $](/files/tex/da6174078cbeae6601684c08526200d9254caa11.png)
![$ (1,0) $](/files/tex/02e6ed02ec9ede67b905b1ca3c64be3eb3c6f11b.png)
![$ t $](/files/tex/4761b031c89840e8cd2cda5b53fbc90c308530f3.png)
![$ (0,1) $](/files/tex/2f2f87361c58fc118cefb1ab5cb288a25e20007f.png)
![$ {\mathbb Z}_n^2 $](/files/tex/784102d74f41429c112d0dd6746a4ab9f1957afe.png)
![$ n-1+t $](/files/tex/26b23f9cc4119b690ed97ea8d21da62ec7899f64.png)
Keywords: subsequence sum; zero sum
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