One thing that has come up a little bit in my work and I believe also e.g. in work of Dmitry Kozlov is such a generalization of the widely used fact (popularized by Gian-Carlo Rota) that the Moebius function $\mu_P(x,y) $ of a poset $P$ may be interpreted as the reduced Euler characteristic of a simplicial complex called the order complex of a subposet of $P$, namely the open interval from $x$ to $y$ (i.e. the subposet comprised of those $z\in P$ satisfying $x < z < y$); the faces in the order complex are the chains of comparable elements in the open interval, so e.g. vertices are individual elements.

There can be situations where one may wish to use a deformation of the usual M"obius function in an inclusion-exclusion counting formula and/or where a variant of the usual M"obius function could have more pleasant formulas, including ones where it is possible to interpret this as the reduced Euler characteristic of the nerve of a small category obtained from a poset by letting each cover relation $u\prec v$ have a multiplicity counting the number of different maps from $u$ to $v$. The deformed M"obius function is the inverse function in the incidence algebra. This hasn't been explored very widely though, at least to my knowledge.

As one example, the usual M"obius function for the poset of set partitions of a multiset gets quite messy, resulting from the fact that it is not multiplicative in terms of decomposition into blocks in a partition; 10 or 15 years ago, Robert Kleinberg and I looked at a multiplicative deformation of it which satisfied much nicer formulas than the traditional M"obius function in this case and interpreted this deformed M"obius function as such a reduced Euler characteristic for the nerve of a small category. Some references in this general direction are:

1. E. Babson and D. Kozlov, Group actions on posets, J. Algebra, 285 (2005), no. 2, 439--450.

2. P. Hersh, Chain decomposition and the flag f-vector, J. Combin. Theory A, 103 (2003), 27--52

3. P. Hersh and R. Kleinberg, A multiplicative deformation for the poset of partitions of a multiset, Communicating Mathematics (special volume in honor of Joe Gallian's 65th birthday), 113--118, Contemp. Math., 479, Amer. Math. Soc., Providence, RI, 2009.

One thing that has come up a little bit in my work and I believe also e.g. in work of Dmitry Feichtner-Kozlov is such an extension of the widely used fact (popularized by Gian-Carlo Rota) that the Moebius function $\mu_P(x,y) $ of a poset $P$ may be interpreted as the reduced Euler characteristic of a simplicial complex called the order complex associated to a subposet of $P$, namely associated to the open interval from $x$ to $y$ (i.e. the subposet comprised of those $z\in P$ satisfying $x < z < y$); the faces in the order complex are the chains of comparable elements in the open interval, so e.g. vertices are individual elements.

There can be situations where one may wish to use a deformation of the usual Moebius function in an inclusion-exclusion counting formula and/or where a variant of the usual Moebius function could have more pleasant formulas, including ones where it is possible to interpret this as the reduced Euler characteristic of the nerve of a small category obtained from a poset by letting each cover relation $u\prec v$ have a multiplicity counting the number of different maps from $u$ to $v$. The deformed Moebius function is the inverse function in the incidence algebra to this weighted incidence relation. This hasn't been explored very widely though, at least to my knowledge. For me this sort of weighted inclusion-exclusion arose naturally in calculating a symmetric function by an inclusion-exclusion of quasisymmetric functions.

As one example, the usual Moebius function for the poset of set partitions of a multiset gets quite messy, resulting from the fact that it is not multiplicative in terms of decomposition into blocks in a partition; 10 or 15 years ago, Robert Kleinberg and I looked at a multiplicative deformation of it which satisfied much nicer formulas than the traditional Moebius function in this case and interpreted this deformed Moebius function as such a reduced Euler characteristic for the nerve of a small category. Some references in this general direction are:

1. E. Babson and D. Kozlov, Group actions on posets, J. Algebra, 285 (2005), no. 2, 439--450.

2. P. Hersh, Chain decomposition and the flag f-vector, J. Combin. Theory Ser. A, 103 (2003), 27--52

3. P. Hersh and R. Kleinberg, A multiplicative deformation of the Moebius function for the poset of partitions of a multiset, Communicating Mathematics (special volume in honor of Joe Gallian's 65th birthday), 113--118, Contemp. Math., 479, Amer. Math. Soc., Providence, RI, 2009.

One thing that has come up a little bit in my work and I believe also e.g. in work of Dmitry Kozlov is such a generalization of the widely used fact that the Moebius function $\mu_P(x,y) $ of a poset $P$ may be interpreted as the reduced Euler characteristic of a simplicial complex called the order complex of a subposet of $P$, namely the open interval from $x$ to $y$ (i.e. the subposet comprised of those $z\in P$ satisfying $x < z < y$); the faces in the order complex are the chains of comparable elements in the open interval, so e.g. vertices are individual elements.

There can be situations where one may wish to use a deformation of the usual M"obius function in an inclusion-exclusion counting formula and/or where a variant of the usual M"obius function could have more pleasant formulas, including ones where it might be possible to interpret this as the reduced Euler characteristic of a small category where poset covering relations $u\prec v$ have multiplicities counting the number of different maps from $u$ to $v$. This hasn't been explored very widely though, at least to my knowledge.

As one example, the usual M"obius function for the poset of set partitions of a multiset gets quite messy, resulting from the fact that it is not multiplicative; 10 or 15 years ago, Robert Kleinberg and I looked at a multiplicative deformation of it which satisfied much nicer formulas and interpreted this as such a reduced Euler characteristic. Some references in this general direction are:

1. E. Babson and D. Kozlov, Group actions on posets, J. Algebra, 285 (2005), no. 2, 439--450.

2. P. Hersh, Chain decomposition and the flag f-vector, J. Combin. Theory A, 103 (2003), 27--52

3. P. Hersh and R. Kleinberg, A multiplicative deformation for the poset of partitions of a multiset, Communicating Mathematics (special volume in honor of Joe Gallian's 65th birthday), 113--118, Contemp. Math., 479, Amer. Math. Soc., Providence, RI, 2009.

One thing that has come up a little bit in my work and I believe also e.g. in work of Dmitry Kozlov is such a generalization of the widely used fact (popularized by Gian-Carlo Rota) that the Moebius function $\mu_P(x,y) $ of a poset $P$ may be interpreted as the reduced Euler characteristic of a simplicial complex called the order complex of a subposet of $P$, namely the open interval from $x$ to $y$ (i.e. the subposet comprised of those $z\in P$ satisfying $x < z < y$); the faces in the order complex are the chains of comparable elements in the open interval, so e.g. vertices are individual elements.

There can be situations where one may wish to use a deformation of the usual M"obius function in an inclusion-exclusion counting formula and/or where a variant of the usual M"obius function could have more pleasant formulas, including ones where it is possible to interpret this as the reduced Euler characteristic of the nerve of a small category obtained from a poset by letting each cover relation $u\prec v$ have a multiplicity counting the number of different maps from $u$ to $v$. The deformed M"obius function is the inverse function in the incidence algebra. This hasn't been explored very widely though, at least to my knowledge.

As one example, the usual M"obius function for the poset of set partitions of a multiset gets quite messy, resulting from the fact that it is not multiplicative in terms of decomposition into blocks in a partition; 10 or 15 years ago, Robert Kleinberg and I looked at a multiplicative deformation of it which satisfied much nicer formulas than the traditional M"obius function in this case and interpreted this deformed M"obius function as such a reduced Euler characteristic for the nerve of a small category. Some references in this general direction are:

1. E. Babson and D. Kozlov, Group actions on posets, J. Algebra, 285 (2005), no. 2, 439--450.

2. P. Hersh, Chain decomposition and the flag f-vector, J. Combin. Theory A, 103 (2003), 27--52

3. P. Hersh and R. Kleinberg, A multiplicative deformation for the poset of partitions of a multiset, Communicating Mathematics (special volume in honor of Joe Gallian's 65th birthday), 113--118, Contemp. Math., 479, Amer. Math. Soc., Providence, RI, 2009.