## Relation Between the Number of Standard Young Tableaux on a Frame and the Number of Young Tableaux on the Frame’s Subordinate/Superordinate Frames

**Point of Post: **In this post we find a relation between the number of standard Young tableaux on a frame and the number of Young tableaux all the subordinate and superordinate frames to .

*Motivation*

As was stated in our last post we can find a very interesting way to calculate the number, , of standard Young tableaux with . In this post we actually prove this claim. The intuitive idea is clear, by construction of Young tableaux we see that if is a Young Tableaux such that is a -frame then the number must lie in a bottom right corner of and then fixing in that position we see that the possible Young tableaux are just the Young tableaux of and thus it makes sense then that is some sort of sum of where is taken over the subordinate frames to . The other theorem which has to do with finding given the values where is taken over the frames superordinate to .

*The Relation Between and for *

We jump right into the claim:

**Theorem: ***Let be a -frame, then*

**Proof: **Let be the set of all Young tableaux with frame . Let be the block with (by this I merely mean the block which contains the value ). Note that since the row is increasing we must have that it is the last in its row since any block to its right would have to have a lower value than . Similarly, is the bottom of its columns for the same reason, and thus . It clearly then follows that if we let for every , be the elements of with then

where the disjointness is obvious since elements from different ‘s have the value of in different boxes. That said, it’s clear that every standard Young tableau in is really just a standard Young tableau of (which we know is really a legitimate -frame) with then just the extra condition that and moreover it’s clear that every standard Young tableau of can be extended to an element of by taking the standard Young tableau, adding in the box at it’s correct point and then filling it in with . It clearly follows then that . Thus, by and our previous characterization of we see that

*The Relation Between and for *

Of course, as usual in algebra, there is a dual notion to the above relation which we dive right into

**Theorem: ***Let be a -frame, then*

**Proof: **We proceed by induction on the size of the frames. For -frames this obvious since the unique -frame is . So, assume that the result is true for every -frame and let be a -frame. Then, using the previous theorem we note that

In this second sum we are summing over subordinates of superordinates of and it’s clear that except the case when the superordinate is the addition of the new row at the bottom of every such subordinaet of a superordinate is really the superordinate of a subordinate of . Said differently

But, since each is a -frame (being subordinate to ) we may apply the induction hypothesis and the first theorem to get

and thus the induction is complete.

**References:**

1. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

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