First for some notation $$ l(\lambda) = \text{ number of parts in a partition } \lambda \vdash n$$

$$ f_{\lambda} = \text{number of standard Young tableau of shape } \lambda\vdash n$$

The number $f_{\lambda}$ is given by the hook length formula and might not acquire a "nice form". Given an integer $k$ consider the problem of computing $$ \tau_{k}(n) = \displaystyle\sum_{\lambda\vdash n \text{, } l(\lambda)\leq k}f_{\lambda}$$

Contrary to expectation, relatively neat closed forms are known for $\tau_{2}(n)$, $\tau_{3}(n)$ and $\tau_{4}(n)$. Gessel link text proved the following $$\tau_{2}(n) = \binom{n}{\lfloor \frac{n}{2} \rfloor}$$ $$\tau_{3}(n) = M_{n}$$ $$\tau_{4}(n) = C_{\lfloor \frac{n+1}{2} \rfloor}C_{\lceil \frac{n+1}{2} \rceil}$$ where $M_{n}$ denotes the n'th Motzkin number link text and $C_{n}$ denotes the n'th Catalan number link text

(Here both sequences are indexed starting 0)

(Aside: Proving the first two identities bijectively is a cute exercise in my opinion.)

As a by-product of my research I obtained the following identity $$ \displaystyle\sum_{\lambda\vdash n, l(\lambda)=5,\lambda_{5}=1}f_{\lambda} = \displaystyle\frac {\lfloor \frac{k+1}{2} \rfloor (\lceil \frac{k+1}{2} \rceil +1)}{k+1}C_{\lfloor \frac{k+1}{2} \rfloor}C_{\lceil \frac{k+1}{2} \rceil} - C_{\lfloor \frac{k}{2} \rfloor +1}C_{\lceil \frac{k}{2} \rceil +1}+M_{k}$$ where the sum on the left runs over all partitions $\lambda$ with length exactly 5 and minimum part $\lambda_{5}$ being 1. Admittedly this is very specific but my question is what is known about sums of the above sort

a) where the minimum part is fixed and so is the length of the partition ?

b) where $l(\lambda) \leq k$ and the k'th part $\lambda_{k} \leq i$ for a fixed non-negative integer $i$?

Gessel, I believe, used some really clever symmetric function manipulation to obtain the identities mentioned earlier. I'd appreciate if somebody has seen this stuff elsewhere ( i.e. reference other than Gessel / Gouyou-Beauchamps) and directs me. Thanks!


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