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Coalgebras appear naturally in combinatorics as describing ways one can decompose objects into other objects of the same type. For example, the coalgebra structure on $k[x]$ given by

$$x^n \mapsto \sum_{k=0}^n {n \choose k} x^k \otimes x^{n-k}$$

describes the ways in which one can decompose a set into two subsets. (Note that the convolution product of linear functionals $k[x] \to k$ can be identified with the product of exponential generating functions, at least in characteristic zero.functions.)

As a more complicated example, there is a coalgebra describing the ways in which one can decompose a connected region of $\mathbb{R}^2$ tiled by finitely many squares into two such regions. There are endless variations on this construction.

I learned this point of view from Gian-Carlo Rota's Coalgebras and bialgebras in combinatorics, which I currently cannot find an online copy of...

Coalgebras appear naturally in combinatorics as describing ways one can decompose objects into other objects of the same type. For example, the coalgebra structure on $k[x]$ given by

$$x^n \mapsto \sum_{k=0}^n {n \choose k} x^k \otimes x^{n-k}$$

describes the ways in which one can decompose a set into two subsets. (Note that the convolution product of linear functionals $k[x] \to k$ can be identified with the product of exponential generating functions, at least in characteristic zero.)

As a more complicated example, there is a coalgebra describing the ways in which one can decompose a connected region of $\mathbb{R}^2$ tiled by finitely many squares into two such regions. There are endless variations on this construction.

I learned this point of view from Gian-Carlo Rota's Coalgebras and bialgebras in combinatorics, which I currently cannot find an online copy of...

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Coalgebras appear naturally in combinatorics as describing ways one can decompose objects into other objects of the same type. For example, the coalgebra structure on $k[x]$ given by

$$x^n \mapsto \sum_{k=0}^n {n \choose k} x^k \otimes x^{n-k}$$

describes the ways in which one can decompose a set into two subsets. As a more complicated example, there is a coalgebra describing the ways in which one can decompose a connected region of $\mathbb{R}^2$ tiled by finitely many squares into two such regions.

I learned this point of view from Gian-Carlo Rota's Coalgebras and bialgebras in combinatorics, which I currently cannot find an online copy of...