A Statistic on Involutions
A Statistic on Involutions
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卷合统计
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
Murali K. Srinivasan
中科院分区：
文献类型：
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作者：
R. S. Deodhar;Murali K. Srinivasan
AbstractWe define a statistic, called weight, on involutions and consider two applications in which this statistic arises. Let I(n) denote the set of all involutions on [n](={1,2,..., n}) and let F(2n) denote the set of all fixed point free involutions on [2n]. For an involution δ, let |δ| denote the number of 2-cycles in δ. Let[ n]q=1+q+⋯+qn-1 and let
$$left( {_{ ext{k}}^{ ext{n}} }
ight)q$$
denote the q-binomial coefficient. There is a statistic wt on I(n) such that the following results are true.(i) We have the expansion
$$left( {_{ ext{k}}^{ ext{n}} }
ight)q = sumlimits_{delta in I(n)} {(q - 1)left| delta
ight|} left( {_{k - left| delta
ight|}^{n - 2left| delta
ight|} }
ight).$$
(ii) An analog of the (strong) Bruhat order on permutations is defined on F(2n) and it is shown that this gives a rank-2
$$(_2^n )$$
graded EL-shellable poset whose order complex triangulates a ball. The rank of δ∈F(2n) is given by wt(δ) and the rank generating function is [1]q[3]q⋯[2n-1]q.
AbstractWe define a statistic, called weight, on involutions and consider two applications in which this statistic arises. Let I(n) denote the set of all involutions on [n](={1,2,..., n}) and let F(2n) denote the set of all fixed point free involutions on [2n]. For an involution δ, let |δ| denote the number of 2-cycles in δ. Let[ n]q=1+q+⋯+qn-1 and let
$$left( {_{ ext{k}}^{ ext{n}} }
ight)q$$
denote the q-binomial coefficient. There is a statistic wt on I(n) such that the following results are true.(i) We have the expansion
$$left( {_{ ext{k}}^{ ext{n}} }
ight)q = sumlimits_{delta in I(n)} {(q - 1)left| delta
ight|} left( {_{k - left| delta
ight|}^{n - 2left| delta
ight|} }
ight).$$
(ii) An analog of the (strong) Bruhat order on permutations is defined on F(2n) and it is shown that this gives a rank-2
$$(_2^n )$$
graded EL-shellable poset whose order complex triangulates a ball. The rank of δ∈F(2n) is given by wt(δ) and the rank generating function is [1]q[3]q⋯[2n-1]q.