2023-07-02

## Preliminaries

We use standard notation, $\setN$ is the set of natural numbers $\{0,1,2,\dotsc,\}$ and $[n]\coloneqq \{1,2,\dotsc,n\}.$

## Partitions

A partition is a weakly decreasing sequence of positive integers, referred to as *parts*.
A partition $\lambda$ can be described as $1^{m_1},2^{m_2},\dotsc,k^{m_k}$
where $m_i$ is the number of parts of $\lambda$ equal to $i.$
The type of $\lambda$ is the vector $(m_1,\dotsc,m_k).$
For example, the partition $444332111111$ can in a more compact form be described as $1^6,2^1,3^2,4^3,$
and it has type $(6,1,2,3).$

Partitions may be illustrated as Young diagrams.

**Example.**

Consider $\lambda=(7,5,5,4,3).$ There are two common ways to draw $\lambda$ as Young diagram, the English way — which is what is mainly used here, and the French way. The English way uses matrix coordinates while the French uses Cartesian coordinates in the first quadrant. The lengths of the rows are determined by the parts of $\lambda.$

$\circ$ | ||||||

$\circ$ | ||||||

$\circ$ | ||||||

$\circ$ | ||||||

There is a third way to describe a partition, using Frobenius coordinates. Let $d$ be the number of boxes on the main diagonal in the Young diagram of $\lambda.$ This is the Durfee size of $\lambda.$ The Frobenius coordinates of $\lambda$ is given by $(a_1,a_2,\dotsc,a_d|b_1,b_2,\dotsc,b_d)$ where $a_i = \lambda_i-i$ is the number of boxes to the right of the diagonal in row $i,$ and $b_i = \lambda'_i-i$ is the number of boxes below the diagonal in column $i.$ In the figure above, $d=4$ and $\lambda=(6,3,2,0|4,3,2,0).$

The conjugate of a partition $\lambda$ is denoted $\lambda',$ and correspond to transposing the (English) Young diagram. Thus, the conjugate of $(7,5,5,4,3)$ is $(5,5,5,4,3,1,1),$ which is simply the sizes of the columns.

### Compositions

A composition of $n$ is a vector with positive integer entries summing to $n.$
Hence, every partition is a composition. These are also sometimes called *strong compositions*.
It is a classical exercise to show that there are $2^{n-1}$ compositions of $n.$

A weak composition of $n$ is a vector with non-negative integer entries summing to $n.$

### Partial orderings on partitions

We write $\lambda \supseteq \mu$ if $\lambda_i \geq \mu_i$ for all $i.$ The dominance order is defined as

\[ \lambda \trianglerighteq \mu \text{ whenever } \lambda_1+\dotsb + \lambda_i \geq \mu_1 + \dotsb + \mu_i \text{ for all $i$}. \]Dominance order is defined in the same manner for compositions and weak compositions.

### Skew shapes and ribbons

If $\lambda \supseteq \mu,$ we let $\lambda/\mu$ denote the set of boxes in the Young diagram of $\lambda$ which do not appear in $\mu.$ We say that $\lambda/\mu$ is a skew shape.

A skew shape is a ribbon if it is connected and does not contain a $2\times 2$-arrangement of boxes. The following figure illustrates a $12$-ribbon, as it has $12$ boxes.

$h$ | |||||||

$t$ |

Ribbons are sometimes called border strips or rim-hooks. Ribbons play a special role in the definition of border-strip tableaux. The head of a ribbon is the rightmost box in the first row of the ribbon, while the tail is the leftmost box in the last row of the ribbon.

See Thm. 10.7.1 in https://arxiv.org/pdf/1503.05930.pdf for the following result.

**Theorem.**

Let $\lambda/\mu$ be a skew shape with $\ell$ rows. The number of partitions $\nu$ satisfying $\mu \subseteq \nu \subseteq \lambda$ is given by the determinant

\[ \left\vert \binom{ \lambda_i - \mu_j + 1 }{ i-j+1 } \right\vert_{1\leq i,j \leq \ell}. \]For non-skew shapes, we can also interpret such partitions $\nu$ as lattice points in Stanley–Pitman polytopes [SP02].

\Per{ I think our reduced lattice pt polynomials are mixed volumes of Stanley–Pitman polytopes.}### Statistics on partitions

Define $\partitionN(\lambda) \coloneqq \sum_i (i-1) \lambda_i = \sum_{i \geq 2} \binom{\lambda'_i}{2}$ and let $z_\lambda \coloneqq \prod_j \left( m_j! j^{m_j} \right)$ where $m_i$ is the number of parts of $\lambda$ of size $i.$

Note that $z_\lambda$ is the number of elements in $\symS_n,$ which fixes $g_\lambda$ under conjugation, where $g_\lambda$ is some fixed permutation of type $\lambda.$ That is,

\[ z_\lambda = |\{ \sigma \in \symS_n : \sigma g_\lambda \sigma^{-1} = g_\lambda \}|. \]The $z_\lambda$ often shows up as a normalizing factor when dealing with power-sum symmetric functions, and in particular, in the Hall inner product.

### Hook lengths and cores

Given a box $(i,j)$ in the Ferrers diagram of $\lambda,$ the hook-length of the box is given by $1 + (\lambda_i-i) + (\lambda'_j-j).$ For example, the hook-length in the diagram below has been written in each box.

$11$ | $10$ | $9$ | $7$ | $5$ | $2$ | $1$ |

$8$ | $7$ | $6$ | $4$ | $2$ | ||

$7$ | $6$ | $5$ | $3$ | $1$ | ||

$5$ | $4$ | $3$ | $1$ | |||

$3$ | $2$ | $1$ |

The content of a box $(i,j)$ is $j-i,$ where $j$ is the column index, and $i$ is the row index.

A $p$-core is a partition where none of the hook-lengths are divisible by $p.$ A closely related concept is the notion of partition quotients.

The hook formula states that

\[ |\SYT(\lambda)| = \frac{n!}{\prod_{\square \in \lambda} h(\square)}, \]where $h(\square)$ denotes the hook length of the box. There is a $q$-analog of this identity.

There is a more complicated formula for skew diagrams, due to H. Naruse [Nar14]. For a proof, see Morales–Pak–Panova [MPP18]. An extension to cylindric diagrams is conjectured in [ST21b].

## Compositions and weak compositions

Similar to partitions, a composition of $n$ is an integer vector of positive integers summing to $n.$ For example,

**Example (Compositions of 4).**

There are $2^{n-1}$ compositions of $n.$ For example,

\[ (4), (3,1), (1,3), (2,2), (2,1,1), (1,2,1), (1,1,2), (1,1,1,1) \]are the compositions of $4.$

A weak composition allows for parts to be equal to $0.$

## Permutations and words

The set of permutations of $n$ elements is denoted $\symS_n.$ Let $\pi \in \symS_n,$ or more generally, a word of length $n.$ The descent set $\DES(\pi)\subseteq [n-1]$ is defined as

\[ \DES(\pi) \coloneqq \{i \in [n-1] : \pi_i \gt \pi_{i+1} \}. \]The set of inversions is the set of pairs,

\[ \INV(\pi) \coloneqq \{(i,j) \in [n]\times [n] : \pi_i \gt \pi_{j} \}. \]We define the major index on permutations and words:

\[ \inv(\pi) \coloneqq |\INV(\pi)| \text{ and } \maj(\pi) \coloneqq \sum_{i \in \DES(\pi)} i. \]For permutations, we also define charge as

\[ \charge(\pi) \coloneqq \maj(\rev(\pi^{-1})) \text{ and } \cocharge(\pi) \coloneqq \binom{n}{2} - \charge(\pi). \]**Lemma.**

We have that

\[ \charge(\pi) = \inv \circ \mathrm{foata} \circ \rev \circ (\cdot)^{-1} \circ \pi, \]where $\mathrm{foata}$ denotes the map sending major index to inversions.

The weight or type of a word $w$ is the composition $\alpha$ such that $\alpha_i$ is the number of entries in $w$ equal to $i.$

A word $w_1,\dotsc,w_n$ is called Yamanouchi if the number of occurrences of $i$ is at least as many as the number of occurrences of $i+1$ in every prefix $w_1,\dotsc,w_k.$

### Bruhat order and weak order

The Bruhat order (or strong order) on permutations is a partial order $\leq$ on $\symS_n.$ We have that $\sigma \leq \tau$ if $\tau$ can obtained from $\sigma$ via a sequence of transpositions, which increase the number of inversions in each step. Alternatively, we have that $\sigma \leq \tau$ if and only if there are reduced words $w_\sigma,$ $w_\tau$ for $\sigma$ and $\tau$ such that $w_\sigma$ is a subword of $w_\tau.$

The Bruhat order is a ranked poset, with rank given by the number of inversions.

The (right) weak order is a subposet of the Bruhat order,
where only *simple transpositions* are allowed, when multiplying on the right.
This corresponds to swapping adjacent entries.
Let us write $\leq_R$ for this relation.
Alternatively, we have that $\sigma \leq_R \tau$ if and only if
there are reduced words $w_\sigma,$ $w_\tau$ for $\sigma$ and $\tau$
such that $w_\sigma$ is a *prefix* of $w_\tau.$

In this figure, we show the Hasse diagram of the Bruhat order for $\symS_4.$ The solid edges indicate covering relations in the (right) weak order. See [Ch. 10.5, Ful97], [Ex. 183, 185, Sta11] and [Ch.2-Ch.3, Bjo05] for more properties and alternative definitions.

### Affine permutations

The set of affine permutations $\symAff_n$ is the set of bijections $\pi: \setZ \to \setZ$ satisfying $\pi(i+n)=\pi(i)+n$ for all $i\in \setZ$ and $\pi(1)+\pi(2)+\dotsb+\pi(n)=1+2+\dotsb+n.$

## Tableaux

A tableau is a filling of the boxes determined by a partition, or sometimes a more general configuration of boxes.

The reading word of a tableau is the word obtained by concatenating the rows, starting from the bottom row in English notation.

The set of standard Young tableaux of shape $\lambda,$ denoted $\SYT(\lambda),$ is the set of bijective fillings $\lambda \to [n],$ such that rows and columns are increasing with row and column index. The set of semistandard Young tableaux, $\SSYT(\lambda),$ is defined as the set of fillings with entries in $\setP,$ with weakly increasing rows and strictly increasing columns. We let $\SSYT(\lambda,\mu)$ denote the (finite) set of semi-standard Young tableaux with shape $\lambda$ and type $\mu,$ that is, there are $\mu_i$ entries equal to $i$ in each such tableau.

**Example.**

Consider the following tableau $T.$

$1$ | $2$ | $5$ | $7$ |

$3$ | $4$ | ||

$6$ |

The reading word of $T$ is $6341257,$ denoted $\rw(T).$ Note that if $T$ is a standard or semistandard Young tableau, then $T$ is uniquely determined by its reading word.

Whenever $T \in \SYT(\lambda),$ we define $\charge(T)$ and $\cocharge(T)$ as the charge and cocharge, respectively, of their reading words.

Given $T \in \SYT(\lambda)$ let

\[ \DES(T) \coloneqq \{i \in [n-1] : i+1 \text{ appears in a lower row than } i \}. \]We then let $\maj(T) \coloneqq \sum_{i \in \DES(T)} i.$ Thus, $\maj(T) = \maj(\rw(T)^{-1}).$

### SYT as Yamanouchi words

The set $\SYT(\lambda)$ is in bijection with Yamanouchi words with weight $\lambda.$ The bijection $f(T)=w$ is given by $w_i = j$ if $i$ occur in row $j$ of $T.$ This map can be quite useful at times. Note that $\maj(T) = \binom{n}{2}-\maj(w).$

### SSYT as lattice paths

The approach to prove the two Jacobi–Trudi identities, by using the Lindström–-Gessel–-Viennot lemmma is to realize that semistandard Young tableaux can be put in correspondence with sets of non-intersecting lattice paths. Let $T \in \SSYT(\lambda/\mu)$ where the entries are bounded by $m.$

*Row bijection*:
Each row $i$ of $T$ is mapped to a lattice path $P_i,$ with steps in the set $\{ (0,1), (1,0)\}.$
The starting vertex of $P_i$ is $(-i + 2\mu_i,1),$
the ending vertex is $(-i+\mu_i+\lambda_i,m),$
and its number of horizontal steps on $y$-coordinate $j$ is given by the number of entries in
row $i$ equal to $j.$ The length of path $P_i$ is $m-1 + \lambda_i-\mu_i.$
This bijection can be used to prove the first Jacobi–Trudi identity.

*Column bijection*:
Each column of $T$ is mapped to a path $P'_i,$
with steps in the set $\{ (-1,1), (1,1)\}.$
The starting vertex of $P'_i$ is given by $(2i - 2\mu'_i, 1),$
the ending vertex is $(2i - 2\lambda'_i + m ,m)$
and the $j$th step of $P'_i$ is $(-1,1)$ if $j$ is present in column $i,$
otherwise it is $(1,1).$ Each path has length $m.$
This bijection can be used to prove the second Jacobi–Trudi identity.

**Example (Bijection to lattice paths).**

Consider the following skew semistandad Young tableau $T.$

$1$ | $1$ | $2$ | $2$ | $3$ | $4$ | $6$ | |||

$1$ | $2$ | $2$ | $3$ | $4$ | $5$ | $5$ | |||

$2$ | $3$ | $4$ | $4$ | $5$ | $6$ | $6$ | |||

$1$ | $1$ | $3$ | $5$ | $5$ | $6$ | ||||

$2$ | $3$ | $4$ | $6$ | ||||||

$4$ | $5$ | $6$ | |||||||

$6$ |

The rows give rise to the following set of non-intersecting lattice paths, where the topmost row in $T$ corresponds to the rightmost path.

The columns in $T$ give rise to the following set of non-intersecting lattice paths, where the leftmost column corresponds to the leftmost path.

The following appears in https://arxiv.org/pdf/2003.04957.pdf, and is proved using lattice paths.

**Theorem.**

Let $A,B$ be subsets of $\{0,1,\dotsc,n\}$ of equal size and let $A^c,$ $B^c$ be their complements. Then

\[ \det \left[ \completeH_{b-a}( x_1,x_2,\dotsc,x_{a+1} ) \right]_{a \in A, b \in B} = \det \left[ \elementaryE_{a'-b'}( x_1,x_2,\dotsc,x_{a'} ) \right]_{a' \in A^c, b' \in B^c}. \]Moreover, this is related to C. A. Aitken's identity,

\[ \det \left[ \completeH_{b-a}(\xvec ) \right]_{a \in A, b \in B} = \det \left[ \elementaryE_{a'-b'}( \xvec ) \right]_{a' \in A^c, b' \in B^c}. \]See also [Ste90] for background on counting lattice paths with pfaffians.

One can also prove the Weyl alternant formula by LGV, see https://arxiv.org/pdf/2003.09215.pdf.

### SSYT as lattice points in a polytope

Semistandard Young tableaux are in bijection with Geltand–Tsetlin patterns, which are lattice points in Gelfand–Tsetlin polytopes.

### The companion map

In [Def. 15, KM19], the authors describes a companion map, which interchanges the role of shape and weight of fillings.

## References

- [Bjo05] Anders Björner. Combinatorics of Coxeter groups. Springer, 2005.
- [Ful97] William Fulton. Young tableaux: with applications to representation theory and geometry. London Mathematical Society Student Texts (Book 35), Cambridge University Press, 1997.
- [KM19] Ryan Kaliszewski and Jennifer Morse. Colorful combinatorics and Macdonald polynomials. European Journal of Combinatorics, 81:354–377, October 2019.
- [MPP18] Alejandro H. Morales, Igor Pak and Greta Panova. Hook formulas for skew shapes I. $q$-analogues and bijections. Journal of Combinatorial Theory, Series A, 154:350–405, February 2018.
- [Nar14] Hiroshi Naruse. Schubert calculus and hook formula. Online 2014. Talk slides at 73rd Sem. Lothar. Combin., Strobl, Austria, 2014
- [SP02] Richard P. Stanley and Jim Pitman. A polytope related to empirical distributions, plane trees, parking functions, and the associahedron. Discrete Computational Geometry, 27(4):603–602, January 2002.
- [ST21b] Takeshi Suzuki and Yoshitaka Toyosawa. On hook formulas for cylindric skew diagrams. arXiv e-prints, 2021.
- [Sta11] Richard P. Stanley. Enumerative Combinatorics: Volume 1. Cambridge University Press, Second edition, 2011.
- [Ste90] John R. Stembridge. Nonintersecting paths, pfaffians, and plane partitions. Advances in Mathematics, 83(1):96–131, September 1990.