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Abstract
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Let K be a field and R = K[x_1, . . . , x_n] be the polynomial ring in n variables over a field K. Let Δ be a simplicial complex on n vertices and I = I_Δ be its Stanley Reisner ideal. In this paper, we show that if I is a matroidal ideal then the following conditions are equivalent: (i) Δ is sequentially Cohen-Macaulay; (ii) Δ is shellable; (iii) Δ is vertex decomposable. Also, if I is a minimally generated by u_1, . . . , u_s such that s<=3 or supp(u_i) \cup supp(u_j) = {x_1, . . . , x_n} for all i ̸= j, then Δ is vertex decompos able. Furthermore, we prove that if I is a monomial ideal of degree 2 then I is weakly polymatroidal if and only if I has linear quotients if and only if I is vertex splittable.
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