[seqfan] bracelets versus Young Tableaux (A000085)

Wouter Meeussen wouter.meeussen at telenet.be
Sat Feb 2 16:31:43 CET 2013


does anybody know of a one-to-one relation between
bracelets of 2n+1 beads with n+1 black ones and n white ones,
and Young Tableaux (or involutions) of the partitions of n?

example for n=4:
tableaux:
{{1, 2, 3, 4}},
{{1, 3, 4}, {2}},
{{1, 2, 4}, {3}},
{{1, 2, 3}, {4}},
{{1, 3}, {2, 4}},
{{1, 2}, {3, 4}},
{{1, 4}, {2}, {3}},
{{1, 3}, {2}, {4}},
{{1, 2}, {3}, {4}},
{{1}, {2}, {3}, {4}}

bracelets:
{0,1,0,1,0,1,0,1,0},
{0,0,0,0,0,1,1,1,1},
{0,0,0,0,1,0,1,1,1},
{0,0,0,0,1,1,0,1,1},
{0,0,0,1,0,0,1,1,1},
{0,0,0,1,0,1,0,1,1},
{0,0,0,1,0,1,1,0,1},
{0,0,0,1,1,0,0,1,1},
{0,0,1,0,0,1,0,1,1},
{0,0,1,0,1,0,0,1,1}

-----------------------------------------
<<Combinatorica`;
Apply[Join, Tableaux /@ Partitions[4]]
and
ListNecklaces[2*4+ 1, {0, 1}, Dihedral]

Wouter.


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