Doubly Stochastic Matrices and Schur-Weyl Duality for Partition Algebras

We prove that the permutations of $\{1,\dots, n\}$ having an increasing (resp., decreasing) subsequence length $n-r$ index a subset set all $r$th Kronecker powers $n \times n$ permutation matrices which is basis for linear span set. Thanks to known Schur-Weyl duality, this gives new centralizer algebra partition acting on tensor power vector space. give some related results doubly stochastic in algebra.