Equivariant differential form

In combinatorial mathematics, the hook-length formula is a formula for the number of standard Young tableaux whose shape is a given Young diagram. It has applications in diverse areas such as representation theory, probability, and algorithm analysis; for example, the problem of longest increasing subsequences.

Definitions and statement

Let $\lambda =(\lambda _{1},\ldots ,\lambda _{m})$ be a partition of $n$. It is customary to interpret $\lambda$ graphically as a Young diagram, namely a left-justified array of square cells with $m$ rows and $\lambda _{i}$ cells in the $i$th row for each $1\leq i\leq m$. A standard Young tableau of shape $\lambda$ is a Young diagram of shape $\lambda$ in which each of the $n$ cells contains a distinct integer between 1 and $n$ (i.e., no repetition), such that each row and each column form increasing sequences. For each cell of the Young diagram in coordinates $(i,j)$ (that is, the cell in the $i$th row and $j$th column), the hook $H_{\lambda }(i,j)$ is the set of cells $(a,b)$ such that $a=i$ and $b\geq j$ or $a\geq i$ and $b=j$. The hook-length $h_{\lambda }(i,j)$ is the number of cells in the hook $H_{\lambda }(i,j)$.

Then the hook-length formula expresses the number of standard Young tableaux of shape $\lambda$, sometimes denoted by $d_{\lambda }$, as

$d_{\lambda }={\frac {n!}{\prod h_{\lambda }(i,j)}},$

where the product is over all cells $(i,j)$ of $\lambda$.

Example

A tableau listing the hook length of each cell in the Young diagram
$(4,3,1,1)$

The figure on the right shows hook-lengths for all cells in the Young diagram Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. of the partition
9 = 4 + 3 + 1 + 1. Then the number of Young tableaux $d_{\lambda }$ for this Young diagram can be computed as

$d_{\lambda }={\frac {9!}{7\cdot 5\cdot 4\cdot 3\cdot 2\cdot 2\cdot 1\cdot 1\cdot 1}}=216.$

History

There are other formulas for $d_{\lambda }$, but the hook-length formula is particularly simple and elegant. The hook-length formula was discovered in 1954 by J. S. Frame, G. de B. Robinson, and R. M. Thrall by improving a less convenient formula expressing $d_{\lambda }$ in terms of a determinant. This earlier formula was deduced independently by G. Frobenius and A. Young in 1900 and 1902 respectively using algebraic methods. P. A. MacMahon found an alternate proof for the Young–Frobenius formula in 1916 using difference methods.

Despite the simplicity of the hook-length formula, the Frame–Robinson–Thrall proof is uninsightful and does not provide an intuitive argument as to why hooks appear in the formula. The search for a short, intuitive explanation befitting such a simple result gave rise to many alternate proofs for the hook-length formula. A. P. Hillman and R. M. Grassl gave the first proof that illuminates the role of hooks in 1976 by proving a special case of the Stanley hook-content formula, which is known to imply the hook-length formula. C. Greene, A. Nijenhuis, and H. S. Wilf found a probabilistic proof using the hook walk in which the hook lengths appear naturally in 1979. J. B. Remmel adapted the original Frame–Robinson–Thrall proof into the first bijective proof for the hook-length formula in 1982. A direct bijective proof was first discovered by D. S. Franzblau and D. Zeilberger in 1982. D. Zeilberger also converted the Greene–Nijenhuis–Wilf hook walk proof into a bijective proof in 1984. A simpler direct bijective proof was announced by Igor Pak and Alexander V. Stoyanovskii in 1992, and its complete proof was presented by the pair and Jean-Christophe Novelli in 1997.

Meanwhile, the hook-length formula has been generalized in several ways. R. M. Thrall found the analogue to the hook-length formula for shifted Young Tableaux in 1952. B. E. Sagan gave a shifted hook walk proof for the hook-length formula for shifted Young tableaux in 1980. B. E. Sagan and Y. N. Yeh proved the hook-length formula for binary trees using the hook walk in 1989.

Proofs

Knuth's heuristic argument

The hook-length formula can be understood intuitively using the following heuristic, but incorrect, argument suggested by D. E. Knuth. Given that each element of a tableau is the smallest in its hook and filling the tableau shape at random, the probability that cell $(i,j)$ will contain the minimum element of the corresponding hook is the reciprocal of the hook length. Multiplying these probabilities over all $i$ and $j$ gives the formula. This argument is fallacious since the events are not independent.

Knuth's argument is however correct for the enumeration of labellings on trees satisfying monotonicity properties analogous to those of a Young tableau. In this case, the 'hook' events in question are in fact independent events.

Probabilistic proof using the hook walk

This is a probabilistic proof found by C. Greene, A. Nijenhuis, and H. S. Wilf in 1979. Here is a sketch of the proof. Define

$e_{\lambda }={\frac {n!}{\prod _{(i,j)\in Y(\lambda )}h_{\lambda }(i,j)}}.$

we would like to show that $d_{\lambda }=e_{\lambda }$. The first observation about $d_{\lambda }$ is

$d_{\lambda }=\sum _{\mu \uparrow \lambda }d_{\mu },$

Corners of the Young diagram (5,3,2,1,1)

where $\mu \uparrow \lambda$ denotes that $\mu$ are Young tableau obtained from $\lambda$ by deleting one corner cell from $\lambda$. The sum is taken over all such $\mu$. Here we are taking the convention that $d_{\phi }=1$, where $\phi$ denotes the empty diagram. The explanation for the above equation is that the maximal entry of the Young tableau of shape $\lambda$ occurs at one of its corner cells. By deleting that cell we will obtain a Young tableau of shape $\mu$. Since the number of Young tableau of shape $\mu$ is $d_{\mu }$, taking the sum over all such $\mu$ we get the equation.

Notice that we also have $e_{\phi }=1$. Therefore it is enough to show that

$e_{\lambda }=\sum _{\mu \uparrow \lambda }e_{\mu },$

and the result $d_{\lambda }=e_{\lambda }$ then follows by induction. The above sum can also be viewed as a sum of probabilities by rewriting the equation to be shown as

$\sum _{\mu \uparrow \lambda }{\frac {e_{\mu }}{e_{\lambda }}}=1.$

We therefore need to show that the numbers ${\frac {e_{\mu }}{e_{\lambda }}}$ define a probability measure on the set of Young diagrams $\mu$ (where $\mu \uparrow \lambda$). This is done in a constructive way by defining a random walk, called the hook walk, on the cells of the Young diagram $\lambda$, which eventually selects one of the corner cells of $\lambda$ (which are in bijection with diagrams $\mu$ for which $\mu \uparrow \lambda$). The hook walk is defined by the following rules.

(1) Pick a cell uniformly at random from $|\lambda |$ cells. Start the random walk from there.

(2) Successor of current cell $(i,j)$ is chosen uniformly at random from the hook $H_{\lambda }(i,j)\setminus \{(i,j)\}$.

(3) Continue until you reach at one of the corner cells, call it ${\textbf {c}}$.

Proposition: For any corner cell $(a,b)$ of $\lambda$ we have

$\mathbb {P} \left({\textbf {c}}=(a,b)\right)={\frac {e_{\mu }}{e_{\lambda }}},$

where $\mu =\lambda \setminus \{(a,b)\}$.

Once we have the above proposition, taking the sum over all possible corner cells ${\textbf {c}}=(a,b)$ we have $\sum _{\mu \uparrow \lambda }{\frac {e_{\mu }}{e_{\lambda }}}=1$, as claimed.

Other proofs

Mention that there are many different proofs, give references. Mention the bijective proofs of Franzblau–Zeilberger and Novelli–Pak–Stoyanovskii and explain why they are important.

Connection to representation theory

The hook-length formula is of great importance in the representation theory of the symmetric group $S_{n}$, where the number $d_{\lambda }$ is known to be equal to the dimension of the irreducible representation $V_{\lambda }$ associated to $\lambda$, and is frequently denoted by $\dim V_{\lambda }$, $\dim \lambda$ or $f^{\lambda }$.

Detailed discussion

The irreducible representations $V_{\lambda }$ of the symmetric group are indexed by partitions $\lambda$ of $n$ (for an explicit construction see Specht module) . Their characters are related to the theory of symmetric functions via the Hall inner product in the following magical formula

$\chi ^{\lambda }(w)=\langle s_{\lambda },p_{\tau (w)}\rangle$

where $s_{\lambda }$ is the schur function associated to $\lambda$ and $p_{\tau (w)}$ is the power symmetric function of the partition $\tau (w)$ associated to the cycle decomposition of $w$. For example, if $w=(154)(238)(6)(79)$ then $\tau (w)=(3,3,2,1)$.

Since $e=(1)(2)\cdots (n)$ in cycle notation, $\tau (e)=1+1+\cdots +1=1^{(n)}$. Then the formula says

$\dim V_{\lambda }=\chi ^{\lambda }(e)=\langle s_{\lambda },p_{1^{(n)}}\rangle$

Considering the expansion of schur functions in terms on monomial symmetric functions using the Kostka numbers

$s_{\lambda }=\sum _{\mu }K_{\lambda \mu }m_{\mu }.\$

the inner product with $p_{1^{(n)}}=h_{1^{(n)}}$ is $K_{\lambda 1^{(n)}}$, because $\langle m_{\mu },h_{\nu }\rangle =\delta _{\mu \nu }$. Note that $K_{\lambda 1^{(n)}}$ is equal to $d_{\lambda }$. Hence

$\dim V_{\lambda }=d_{\lambda }.$

An immediate consequence of this is

$\sum _{\lambda \vdash n}\left(f^{\lambda }\right)^{2}=n!$

The above equality also a simple consequence of the Robinson–Schensted–Knuth correspondence.

The computation also shows that:

$(x_{1}+x_{2}+\cdots +x_{k})^{n}=\sum _{\lambda \vdash n}s_{\lambda }f^{\lambda }.$

Which is the expansion of $p_{1^{(n)}}$ in terms of schur functions using the coefficients given by the inner product, because $\langle s_{\mu },s_{\nu }\rangle =\delta _{\mu \nu }$. The above equality can be proven also checking the coefficients of each monomial at both sides and using the Robinson–Schensted–Knuth correspondence or, more conceptually, looking at the decomposition of $V^{\otimes n}$ by irreducible $GL(V)$ modules, and taking characters. See Schur–Weyl duality.

Outline of the proof of the hook formula using Frobenius formula

By the above considerations

$p_{1^{(n)}}=\sum _{\lambda \vdash n}s_{\lambda }f^{\lambda }$

So that

$\Delta (x)p_{1^{(n)}}=\sum _{\lambda \vdash n}\Delta (x)s_{\lambda }f^{\lambda }$

where $\Delta (x)=\prod _{i<j}(x_{i}-x_{j})$ is the Vandermonde determinant

For a given partition $\lambda =(\lambda _{1},\lambda _{2},\cdots ,\lambda _{k})$ define $l_{i}=\lambda _{i}+k-1$ for $i=1,2,\cdots ,k$. For the following we need at least as many variables as rows in the partition, so for now on we work with $n$ variables $x_{1},\cdots ,x_{n}$.

Each term $\Delta (x)s_{\lambda }$ is equal to sums of

$a_{(\lambda _{1}+n-1,\lambda _{2}+n-2,\dots ,\lambda _{k})}(x_{1},x_{2},\dots ,x_{k})=\det \left[{\begin{matrix}x_{1}^{l_{1}}&x_{2}^{l_{1}}&\dots &x_{k}^{l_{1}}\\x_{1}^{l_{2}}&x_{2}^{l_{2}}&\dots &x_{k}^{l_{2}}\\\vdots &\vdots &\ddots &\vdots \\x_{1}^{l_{k}}&x_{2}^{l_{k}}&\dots &x_{k}^{l_{k}}\end{matrix}}\right]$

See Schur function. Since the vector $(l_{1},l_{2},\cdots ,l_{k})$ is different for each partition, this means that the coefficient of $x_{1}^{l_{1}}\cdots x_{k}^{l_{k}}$ in $\Delta (x)p_{1^{(n)}}$, denoted $\left[\Delta (x)p_{1^{(n)}}\right]_{l_{1},\cdots ,l_{k}}$, is equal to $f^{\lambda }$. This is known as the Frobenius Character Formula, which gives one of the earliest proofs All that remains is tracking that coefficient with a mixture of cleverness and brute force:

$\Delta (x)=\sum _{w\in S_{n}}\operatorname {sgn}(w)x_{1}^{w(1)-1}x_{2}^{w(2)-1}\cdots x_{k}^{w(k)-1}$

$(x_{1}+x_{2}+\cdots +x_{k})^{n}=\sum {\frac {n!}{d_{1}!d_{2}!\cdots d_{k}!}}x_{1}^{d_{1}}x_{2}^{d_{2}}\cdots x_{k}^{d_{k}}$

By pairing, the coefficient that we are looking for is

$\sum _{w\in S_{n}}\operatorname {sgn}(w){\frac {n!}{(l_{1}-w(1)+1)!(l_{2}-w(2)+1)!\cdots (l_{k}-w(k)+1)!}}$

which can be written as

${\frac {n!}{l_{1}!l_{2}!\cdots l_{k}!}}\sum _{w\in S_{n}}\operatorname {sgn}(w)\left[(l_{1})(l_{1}-1)\cdots (l_{1}-w(1)+2)\right]\left[(l_{2})(l_{2}-1)\cdots (l_{2}-w(2)+2)\right]\left[(l_{k})(l_{k}-1)\cdots (l_{k}-w(k)+2)\right]$

the sum is equal to the following determinant

$\det \left[{\begin{matrix}1&l_{1}&l_{1}(l_{1}-1)&\dots &\prod _{i=0}^{k-2}l_{1}-i\\1&l_{2}&l_{1}(l_{2}-1)&\dots &\prod _{i=0}^{k-2}l_{2}-i\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&l_{k}&l_{k}(l_{k}-1)&\dots &\prod _{i=0}^{k-2}l_{k}-i\end{matrix}}\right]$

which column reduces to Vandermonde, and we obtain the formula

${\frac {n!}{l_{1}!l_{2}!\cdots l_{k}!}}\prod _{i<j}(l_{i}-l_{j})$

Note that $l_{i}$ is the hook length of the first box in each row of the Young Diagram. From here its easy to clean the mess of the cancellations and get the hook formula.

Connection to longest increasing subsequences

The hook length formula also has important applications to the analysis of longest increasing subsequences in random permutations. If $\sigma _{n}$ denotes a uniformly random permutation of order $n$, $L(\sigma _{n})$ denotes the maximal length of an increasing subsequence of $\sigma _{n}$, and $\ell _{n}$ denotes the expected (average) value of $L(\sigma _{n})$, Anatoly Vershik and Sergei Kerov and independently Benjamin F. Logan and Lawrence A. Shepp showed that when $n$ is large, $\ell _{n}$ is approximately equal to $2{\sqrt {n}}$. This answers a question originally posed by Stanislaw Ulam. The proof is based on translating the question via the Robinson–Schensted correspondence to a problem about the limiting shape of a random Young tableau chosen according to Plancherel measure. Since the definition of Plancherel measure involves the quantity $d_{\lambda }$, the hook length formula can then be used to perform an asymptotic analysis of the limit shape and thereby also answer the original question.

The ideas of Vershik–Kerov and Logan–Shepp were later refined by Jinho Baik, Percy Deift and Kurt Johansson, who were able to achieve a much more precise analysis of the limiting behavior of the maximal increasing subsequence length, proving an important result now known as the Baik–Deift–Johansson theorem. Their analysis again makes crucial use of the fact that $d_{\lambda }$ has a number of good formulas, although instead of the hook length formula it made use of one of the determinantal expressions.

Related formulas

The formula for the number of Young tableau of a given shape was originally derived from the Frobenius determinant formula in connection to representation theory. If the shape of a Young diagram is given by the row lengths $n_{1},\dots ,n_{m}$, then the number of tableau with that shape is given by

$f(n_{1},n_{2},\ldots ,n_{m})={\frac {n!\,\Delta (n_{m},n_{m-1}+1,\ldots ,n_{1}+m-1)}{n_{m}!(n_{m-1}+1)!\cdots (n_{1}+m-1)!}}$

Hook lengths can also be used to give a product representation to the generating function for the number of reverse plane partitions of a given shape. If Template:Mvar is a partition of some integer Template:Mvar, a reverse plane partition of Template:Mvar with shape Template:Mvar is obtained by filling in the boxes in the Young diagram with non-negative integers such that the entries add to Template:Mvar and are non-decreasing along each row and down each column. The hook lengths $h_{1},\dots ,h_{p}$ can be defined as with Young tableau. If Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. denotes the number of reverse plane partitions of Template:Mvar with shape Template:Mvar, then the generating function can be written as

$\sum _{n=0}^{\infty }\pi _{n}x^{n}=\prod _{k=1}^{p}(1-x^{h_{k}})^{-1}$

Stanley discovered another formula for the same generating function. In general, if $A$ is any poset with $n$ elements, the generating function for reverse $A$-partitions is

${\frac {P(x)}{(1-x)(1-x^{2})\cdots (1-x^{n})}}$

where $P(x)$ is a polynomial such that $P(1)$ is the number of natural labelings of $A$.

In the case of a partition $\lambda$, we are considering the poset in its cells given by the relation

$(i,j)\leq (i',j')\iff i\leq i'\qquad {\textrm {and}}\qquad j\leq j'$.

So a natural labeling is simply a standard Young tableau, i.e. $P(1)=f^{\lambda }$

Yet another proof of the hook length formula

Combining the two formulas for the generating functions we have

${\frac {P(x)}{(1-x)(1-x^{2})\cdots (1-x)^{n}}}=\prod _{(i,j)\in \lambda }(1-x^{h_{(i,j)}})^{-1}$

Both sides converge inside the disk of radius one and the following expression makes sense for $|x|<1$

$P(x)={\frac {\prod _{k=1}^{n}(1-x^{k})}{\prod _{(i,j)\in \lambda }(1-x^{h_{(i,j)}})}}.$

It would be violent to plug in 1, but the right hand side is a continuous function inside the unit disk and a polynomial is continuous everywhere so at least we can say

$P(1)=\lim _{x\to 1}{\frac {\prod _{k=1}^{n}(1-x^{k})}{\prod _{(i,j)\in \lambda }(1-x^{h_{(i,j)}})}}.$

Finally, applying L'Hospital's rule $n$ times yields the hook length formula

$P(1)={\frac {n!}{\prod _{(i,j)\in \lambda }h_{(i,j)}}}.$

Specialization of schur functions

Specializing the schur functions to the variables $1,t,t^{2},t^{3},\cdots$ there is the formula

$s_{\lambda }(1,t,t^{2},\cdots )={\frac {t^{n\left(\lambda \right)}}{\prod _{(i,j)\in Y(\lambda )}(1-t^{h_{\lambda }(i,j)})}}$

The number $n(\lambda )$ is defined as

$n(\lambda )=\sum _{i}(i-1)\lambda _{i}=\sum _{i}{\binom {\lambda _{i}'}{2}}$

where $\lambda '$ is the conjugate partition

A formula for the Catalan numbers

The Catalan numbers are ubiquitous in enumerative combinatorics. Not surprisingly, they are also part of this story:

$C_{n}=f^{(n,n)}$

Lets briefly mention why. When doing a Dyck path we may go up (U) or down (D). So for any Dyck path of length $n$ consider the tableaux of shape $(n,n)$ such that the first row is given by the numbers $i$ such that the $i$-th step was up and in the second row given by the positions in which it goes down. For example UUDDUD correspond to the tableaux with rows 125 and 346.

The hook formula gives another way of getting a closed formula for the Catalan numbers

$C_{n}={\frac {(2n)!}{(n+1)(n)\cdots (3)(2)(n)(n-1)\cdots (2)(1)}}={\frac {(2n)!}{(n+1)!n!}}={\frac {1}{n+1}}{\binom {2n}{n}}$

(\TO DO: in the Catalan article they mention this relation and the have a link for hook length formula, which redirects to Young tableau, which is well written but offers no intuition about the math. Those articles should be redirected to this one)

See also

• Young tableau

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• An online book draft on longest increasing subsequences by Dan Romik. Contains discussions of the hook length formula and several of its variants, with applications to the mathematics of longest increasing subsequences.