The bijective mapping between a composition into six parts , [c[1], c[2], c[3], c[4], c[5], c[6]], of n, say (into non-neg. integers) \ and Condorcet vote-count profile with the 1->2->3->1 and 2n+3 voters, i\ s [c[1] + c[4] + c[6] + 1, c[2], c[3], c[2] + c[4] + c[5] + 1, c[3] + c[5] + c[6] + 1, c[1]] and in Maple format [c[1]+c[4]+c[6]+1, c[2], c[3], c[2]+c[4]+c[5]+1, c[3]+c[5]+c[6]+1, c[1]] the sum of the components is 2 c[1] + 2 c[4] + 2 c[6] + 3 + 2 c[2] + 2 c[3] + 2 c[5] as you can see, it is indeed 2n+3 The Condorcet inequlity conditions for 1->2->3->1 are [2 c[6] + 1, 2 c[4] + 1, 2 c[5] + 1] As you can see they are all strictly positive if, c[1], c[2], c[3], c[4], c[5], c[6], are all non-negative (as they should be). Let , [c[123], c[132], c[213], c[231], c[312], c[321]], be a composition (wi\ th non-neg. integers) that constitutes a Condorcet 1->2->3->1 vote-count\ profile with 2n+3 voters The inverse mapping to compositions of n into six non-negative integers is [c[321], c[132], c[213], -1/2 c[132] + 1/2 c[213] - 1/2 c[321] - 1/2 - 1/2 c[312] + 1/2 c[231] + 1/2 c[123], -1/2 c[213] + 1/2 c[321] - 1/2 c[132] - 1/2 + 1/2 c[312] + 1/2 c[231] - 1/2 c[123], -1/2 c[321] + 1/2 c[132] - 1/2 c[213] - 1/2 + 1/2 c[312] - 1/2 c[231] + 1/2 c[123]] and in Maple format [c[321], c[132], c[213], -1/2*c[132]+1/2*c[213]-1/2*c[321]-1/2-1/2*c[312]+1/2*c [231]+1/2*c[123], -1/2*c[213]+1/2*c[321]-1/2*c[132]-1/2+1/2*c[312]+1/2*c[231]-1 /2*c[123], -1/2*c[321]+1/2*c[132]-1/2*c[213]-1/2+1/2*c[312]-1/2*c[231]+1/2*c[ 123]] These mappings are inverses of each other. Indeed applying CondToComp(CompTo\ Cond) `To `, [c[1], c[2], c[3], c[4], c[5], c[6]] gives [c[1], c[2], c[3], c[4], c[5], c[6]] In the other direction, applying CompToCond(CondToComp) to [c[123], c[132], c[213], c[231], c[312], c[321]] Gives [c[123], c[132], c[213], c[231], c[312], c[321]]