SemistandardTableaux

Semistandard Young tableaux of $n$ cells with entries in $\{1, …, k\}$ , summed over every shape $\lambda \vdash n$ , as a lazy indexed family.

SemistandardTableaux(size, max_entry)the SSYT of size cells over every partition shape, entries from 1 to max_entry.

Typed signatures

Type	Package	Applies
`(integer<0..>, integer<0..>) -> indexed_collection<list<list<integer>>>`	`combinatorics`	always

Domain: Collections

Details

A lazy indexed collection; the count is the hook-content formula $s_\lambda(1^k) = \prod_{(r,c) \in \lambda} \frac{k + c - r}{hook(r,c)}$ , summed over every shape $\lambda \vdash n$ — exact and closed-form.
Each element is the filling's rows: weakly increasing left to right, strictly increasing top to bottom — semistandard, not standard, so entries may repeat within a row (unlike a standard Young tableau).
Unranked in shape-then-entries order: by row-length shape first, then the flattened filling.

Enumeration

1 more kept as data, checked against the oracles

Basic #

$Count(SemistandardTableaux(4, 3)) = 39$ : every SSYT of 4 cells with entries in $\{1,2,3\}$ , summed over shape

\mathrm{Count}(\mathrm{SemistandardTableaux}(4, 3))

Out

39

Scope #

The first SSYT of 3 cells with entries in $\{1, 2\}$ , as a list of rows

\mathrm{SemistandardTableaux}(3, 2)[1]

Out

\bigl\lbrack\bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack2\bigr\rbrack\bigr\rbrack

About

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