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Combinatorial proof of Hamiltonian paths on the rook graph

Number of duplicate cases in graphs - hamiltonian and nonhamiltonian pathsCan more than one hamiltonian graph have the same set of hamiltonian paths?Hamiltonian paths in cubic graphRook tour on the chess boardMinimum number of Hamiltonian paths in a strongly connected tournament on $n$ nodesProve that the line graph of a Hamiltonian simple graph is Hamiltonian.Hamiltonian paths in a simple graphHamiltonian cycles and paths in a graphHamiltonian paths in graphHamiltonian paths and cycles of rook graph on $ntimes2$ chessboard

1

$begingroup$

We can be sure that number of Hamiltonian paths on the rook graph for any single cell on $ntimes2$ chessboard equals
$$
H(n+1) = sum_{k=0}^{n}
binom{n}{k}
binom{k}{lfloor{frac{k}{2}rfloor}}
left(n-lfloor{frac{k}{2}rfloor}right)!
left(n-lfloor{frac{k+1}{2}rfloor}right)!$$
Is there a combinatorial proof for it?

graph-theory hamiltonian-path combinatorial-proofs chessboard

share|cite|improve this question

edited Mar 11 at 19:52

gt6989b

34.9k22557

asked Mar 11 at 19:48

user514787user514787

749310

$endgroup$

add a comment | 

1

$begingroup$

We can be sure that number of Hamiltonian paths on the rook graph for any single cell on $ntimes2$ chessboard equals
$$
H(n+1) = sum_{k=0}^{n}
binom{n}{k}
binom{k}{lfloor{frac{k}{2}rfloor}}
left(n-lfloor{frac{k}{2}rfloor}right)!
left(n-lfloor{frac{k+1}{2}rfloor}right)!$$
Is there a combinatorial proof for it?

graph-theory hamiltonian-path combinatorial-proofs chessboard

share|cite|improve this question

edited Mar 11 at 19:52

gt6989b

34.9k22557

asked Mar 11 at 19:48

user514787user514787

749310

$endgroup$

add a comment | 

$begingroup$

We can be sure that number of Hamiltonian paths on the rook graph for any single cell on $ntimes2$ chessboard equals
$$
H(n+1) = sum_{k=0}^{n}
binom{n}{k}
binom{k}{lfloor{frac{k}{2}rfloor}}
left(n-lfloor{frac{k}{2}rfloor}right)!
left(n-lfloor{frac{k+1}{2}rfloor}right)!$$
Is there a combinatorial proof for it?

graph-theory hamiltonian-path combinatorial-proofs chessboard

share|cite|improve this question

edited Mar 11 at 19:52

gt6989b

34.9k22557

asked Mar 11 at 19:48

user514787user514787

749310

$endgroup$

share|cite|improve this question

edited Mar 11 at 19:52

gt6989b

34.9k22557

asked Mar 11 at 19:48

user514787user514787

749310

share|cite|improve this question

edited Mar 11 at 19:52

gt6989b

34.9k22557

asked Mar 11 at 19:48

user514787user514787

749310

edited Mar 11 at 19:52

gt6989b

34.9k22557

edited Mar 11 at 19:52

gt6989b

34.9k22557

asked Mar 11 at 19:48

user514787user514787

749310

asked Mar 11 at 19:48

user514787user514787

749310