Geodesic
A combinatorial proof of a formula for Betti numbers of a stacked polytope
The electronic journal of combinatorics, Tome 17 (2010)
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For a simplicial complex $\Delta$, the graded Betti number $\beta_{i,j}({\bf k}[\Delta])$ of the Stanley-Reisner ring ${\bf k}[\Delta]$ over a field ${\bf k}$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $\Delta$ is the boundary complex of a $d$-dimensional stacked polytope with $n$ vertices for $d\geq3$, then $\beta_{k-1,k}({\bf k}[\Delta])=(k-1){n-d\choose k}$. We prove this combinatorially.
DOI : 10.37236/281
Classification : 05A15, 05E40, 05E45, 52B05 
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     author = {Suyoung Choi and Jang Soo Kim},
     title = {A combinatorial proof of a formula for {Betti} numbers of a stacked polytope},
     journal = {The electronic journal of combinatorics},
     year = {2010},
     volume = {17},
     doi = {10.37236/281},
     zbl = {1215.05011},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/281/}
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Suyoung Choi; Jang Soo Kim. A combinatorial proof of a formula for Betti numbers of a stacked polytope. The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/281

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