In the [Young's lattice](https://en.wikipedia.org/wiki/Young%27s_lattice), the number of branches that connect the $N$'th layer to the $N+1$'th layer has the sequence:
$$
1,2, 4, 7, 12, 19, 30, 45, 67, 97, 139, \cdots
$$
Looking this up on OEIS, leads to [this result](https://oeis.org/A000070). This is nothing but the cumulative sum of integer partitions
$$ \sum\_{k=0}^N p(k)
$$
My question is: why do we expect this number to count the number of branches between two layers of the Young's lattice?



Number of branches between two layers of the Young's lattice
Contra what I originally thought, I’m not sure this is a general fact for all differential posets.


Nonetheless in the case of Young’s lattice it is easy to see this directly from the fact that if an element has $x$ edges coming in from below, it has $x+1$ going out above: we can always add a box in one more position than we can take away a box, in any Young diagram.


