Some comments on "On the generating function for intervals in Young's lattice" by Azam and Richmond

Fuente: arXiv

Saved in:

Bibliographic Details
Main Author:	Snellman, Jan
Format:	Preprint
Published:	2024
Subjects:	Combinatorics 05A17
Online Access:	Acceder al recurso
Tags:	Add Tag No Tags, Be the first to tag this record!

A visual approach to symmetric chain decompositions of finite Young lattices
by: Coggins, Terrance, et al.
Published: (2024)

Complexity in Young's Lattice
by: Wires, Alexander
Published: (2019)

Some elementary remarks on the powers of a partial theta function and corresponding q-analogs of the binomial coefficients
by: Cigler, Johann
Published: (2024)

On a Pieri-like rule for the Petrie symmetric functions
by: Jin, Emma Yu, et al.
Published: (2024)

Bijections and congruences involving lattice paths and integer compositions
by: Dastidar, Manosij Ghosh, et al.
Published: (2024)

A Geometric Characterization of Maximal Unrefinable Partitions via the Keith-Nath Transformation and Young Diagrams
by: Aragona, Riccardo, et al.
Published: (2026)

Truncated theta series from the Bailey lattice
by: Ding, Xiangyu, et al.
Published: (2024)

Combinatorial Proofs of Some Results of Andrews and El Bachraoui
by: Mahanta, Pankaj Jyoti, et al.
Published: (2024)

Palindromicity of the numerator of a statistical generating function
by: Bourn, Rebecca, et al.
Published: (2023)

Ramanujan's partition generating functions modulo $\ell$
by: Bringmann, Kathrin, et al.
Published: (2025)

Laplacians on shifted multicomplexes
by: Snellman, Jan
Published: (2006)

Remarks on MacMahon's $q$-series
by: Ono, Ken, et al.
Published: (2024)

Palindrome Partitions and the Calkin-Wilf Tree
by: Hemmer, David J., et al.
Published: (2024)

Cycles in Austrian Solitaire
by: Mummert, Philip
Published: (2023)

The change-making problem for six coin values and beyond
by: Van Cott, Cornelia A., et al.
Published: (2023)

Transition matrices and Pieri-type rules for polysymmetric functions
by: Khanna, Aditya, et al.
Published: (2024)

Sum of squares of hook lengths and contents
by: Menon, Krishna
Published: (2024)

On $\ell$-regular partitions and Hickerson's identity
by: Liu, Ji-Cai
Published: (2024)

On Blecher and Knopfmacher's Fixed Points for Integer Partitions
by: Hopkins, Brian, et al.
Published: (2023)

Proof of a Conjecture of Nath and Sellers on Simultaneous Core Partitions
by: Sha, Yetong, et al.
Published: (2023)

Blocking Ideals: a method for filtering linear extensions of a finite poset
by: Jaldevik, Albin, et al.
Published: (2025)

Partition division maps, symmetric functions and positivity
by: Alexandersson, Per, et al.
Published: (2026)

Finding systems of functional equations for Andrews-Gordon type series
by: Kılıç, Yalçın Can, et al.
Published: (2025)

Polynomial Expressions for the Dimensions of the Representations of Symmetric Groups and Restricted Standard Young Tableaux
by: Cohen, Avichai, et al.
Published: (2024)

A combinatorial interpretation for certain plethysm and Kronecker coefficients
by: Pak, Igor, et al.
Published: (2025)

A recursive method for the oddness of the number of set-valued tableaux
by: Fujii, Taikei, et al.
Published: (2023)

On partitions associated with elementary symmetric polynomials
by: Ballantine, Cristina, et al.
Published: (2025)

Reciprocals of false theta functions
by: Keith, William J.
Published: (2024)

The Polytope of Probability Functions on a Finite Poset
by: Snellman, Jan
Published: (2025)

Special values of $K$-theoretic Schur $P$- and $Q$-functions
by: Nobukawa, Takahiko, et al.
Published: (2024)

The combinatorics of identities involving overpartitions with distinct parts
by: Li, Haijun
Published: (2025)

A combinatorial proof of Buryak-Feigin-Nakajima
by: Vidalis, Eve
Published: (2022)

Jump and Gradient Invariants in the Partition Graph
by: Lyudogovskiy, Fedor B.
Published: (2026)

On the strong metric dimension of the zero-divisor graph of a lattice
by: Gadge, Pravin, et al.
Published: (2024)

Set-Valued Catalan Combinatorics
by: Lazar, Alexander, et al.
Published: (2024)

On the strong metric dimension of the complement of the zero-divisor graph of a lattice
by: Gadge, Pravin, et al.
Published: (2025)

Partition functions that repel perfect-powers
by: Ono, Ken
Published: (2025)

Tilings of Benzels via the Abacus Bijection
by: Defant, Colin, et al.
Published: (2022)

2-covers of wide Young diagrams
by: Aharoni, Ron, et al.
Published: (2023)

if (filemtime($filename) + $maxlifetime < time()) { unlink($filename); $count++; } } return $count; } /** * A function that is called internally when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ protected function saveSession($sessId, $data): bool { $sessFile = $this->path . '/sess_' . $sessId; if ($handle = fopen($sessFile, 'w')) { $return = false; // Lock the file for exclusive access to avoid issues with multiple // processes writing session simultaneously: if (flock($handle, LOCK_EX)) { $return = fwrite($handle, $data); // Make sure that there's no trailing data by truncating the file to // the correct length: ftruncate($handle, strlen($data)); fflush($handle); flock($handle, LOCK_UN); } fclose($handle); if ($return !== false) { return true; } } // If we got this far, something went wrong with the file output... // It is tempting to throw an exception here, but this code is called // outside of the context of exception handling, so all we can do is // echo a message.

// Anecdotal testing Today and Yesterday seems to indicate destroy() // is called by the garbage collector and everything is good. // Something to keep in mind though. return 0; } /** * Write function that is called when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ public function write($sessId, $data): bool { if ($this->writesDisabled) { return true; } return $this->saveSession($sessId, $data); } /** * A function that is called internally when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ abstract protected function saveSession($sessId, $data): bool; }

* * @return void */ public function writeClose() { // The assumption is that we're using PHP's ext/session. // session_write_close() will actually overwrite $_SESSION with an // empty array on completion -- which leads to a mismatch between what // is in the storage object and $_SESSION. To get around this, we // temporarily reset $_SESSION to an array, and then re-link it to // the storage object. // // Additionally, while you _can_ write to $_SESSION following a // session_write_close() operation, no changes made to it will be // flushed to the session handler. As such, we now mark the storage // object isImmutable. $storage = $this->getStorage(); if (! $storage->isImmutable()) { $_SESSION = $storage->toArray(true); session_write_close(); $storage->fromArray($_SESSION); $storage->markImmutable(); } } /** * Attempt to set the session name * * If the session has already been started, or if the name provided fails * validation, an exception will be raised. * * @param string $name * @return SessionManager * @throws Exception\InvalidArgumentException */ public function setName($name) { if ($this->sessionExists()) { throw new Exception\InvalidArgumentException( 'Cannot set session name after a session has already started'

Environment & details:

GET Data empty

POST Data empty

Files empty

Key	Value
VUFIND_SESSION	"5cef0c542867b646388bfb9e0a54cc03"
ui	"standard"
language	"en"

Session

Key	Value
__Laminas	array:2 [`"_REQUEST_ACCESS_TIME" => 1790731366.7936 "_VALID" => array:1 [ "Laminas\Session\Validator\Id" => "5cef0c542867b646388bfb9e0a54cc03" ]`]
SessionState	Laminas\Stdlib\ArrayObject {#938}
FlashMessenger	Laminas\Stdlib\ArrayObject {#1433}
SessionHelper	Laminas\Stdlib\ArrayObject {#1759}

Server/Request Data

Key	Value
REDIRECT_VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
REDIRECT_HTTPS	"on"
REDIRECT_SSL_TLS_SNI	"info.umar.mx"
REDIRECT_STATUS	"200"
VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
HTTPS	"on"
SSL_TLS_SNI	"info.umar.mx"
HTTP_ACCEPT	"/"
HTTP_USER_AGENT	"Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)"
HTTP_COOKIE	"VUFIND_SESSION=5cef0c542867b646388bfb9e0a54cc03; ui=standard; language=en"
HTTP_ACCEPT_ENCODING	"gzip, br, zstd, deflate"
HTTP_HOST	"info.umar.mx"
PATH	"/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/snap/bin"
SERVER_SIGNATURE	"<address>Apache/2.4.66 (Ubuntu) Server at info.umar.mx Port 443</address>\n"
SERVER_SOFTWARE	"Apache/2.4.66 (Ubuntu)"
SERVER_NAME	"info.umar.mx"
SERVER_ADDR	"200.23.222.19"
SERVER_PORT	"443"
REMOTE_ADDR	"216.73.216.147"
DOCUMENT_ROOT	"/usr/local/vufind/public"
REQUEST_SCHEME	"https"
CONTEXT_PREFIX	""
CONTEXT_DOCUMENT_ROOT	"/usr/local/vufind/public"
SERVER_ADMIN	"admin@localhost"
SCRIPT_FILENAME	"/usr/local/vufind/public/index.php"
REMOTE_PORT	"19836"
REDIRECT_URL	""" /Record/\n arxiv_https___arxiv_org_abs_2401_04030/Similar """
GATEWAY_INTERFACE	"CGI/1.1"
SERVER_PROTOCOL	"HTTP/1.1"
REQUEST_METHOD	"GET"
QUERY_STRING	""
REQUEST_URI	"/Record/%0Aarxiv_https___arxiv_org_abs_2401_04030/Similar"
SCRIPT_NAME	"/index.php"
PHP_SELF	"/index.php"
REQUEST_TIME_FLOAT	1790731366.7892
REQUEST_TIME	1790731366

Environment Variables empty

Registered Handlers

0. Whoops\Handler\PrettyPageHandler

Environment & details:

GET Data empty

POST Data empty

Files empty

Key	Value
VUFIND_SESSION	"5cef0c542867b646388bfb9e0a54cc03"
ui	"standard"
language	"en"

Session

Key	Value
__Laminas	array:2 [`"_REQUEST_ACCESS_TIME" => 1790731366.7936 "_VALID" => array:1 [ "Laminas\Session\Validator\Id" => "5cef0c542867b646388bfb9e0a54cc03" ]`]
SessionState	Laminas\Stdlib\ArrayObject {#938}
FlashMessenger	Laminas\Stdlib\ArrayObject {#1433}
SessionHelper	Laminas\Stdlib\ArrayObject {#1759}

Server/Request Data

Key	Value
REDIRECT_VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
REDIRECT_HTTPS	"on"
REDIRECT_SSL_TLS_SNI	"info.umar.mx"
REDIRECT_STATUS	"200"
VUFIND_LOCAL_DIR	"/usr/local/vufind/local"
HTTPS	"on"
SSL_TLS_SNI	"info.umar.mx"
HTTP_ACCEPT	"/"
HTTP_USER_AGENT	"Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)"
HTTP_COOKIE	"VUFIND_SESSION=5cef0c542867b646388bfb9e0a54cc03; ui=standard; language=en"
HTTP_ACCEPT_ENCODING	"gzip, br, zstd, deflate"
HTTP_HOST	"info.umar.mx"
PATH	"/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/snap/bin"
SERVER_SIGNATURE	"<address>Apache/2.4.66 (Ubuntu) Server at info.umar.mx Port 443</address>\n"
SERVER_SOFTWARE	"Apache/2.4.66 (Ubuntu)"
SERVER_NAME	"info.umar.mx"
SERVER_ADDR	"200.23.222.19"
SERVER_PORT	"443"
REMOTE_ADDR	"216.73.216.147"
DOCUMENT_ROOT	"/usr/local/vufind/public"
REQUEST_SCHEME	"https"
CONTEXT_PREFIX	""
CONTEXT_DOCUMENT_ROOT	"/usr/local/vufind/public"
SERVER_ADMIN	"admin@localhost"
SCRIPT_FILENAME	"/usr/local/vufind/public/index.php"
REMOTE_PORT	"19836"
REDIRECT_URL	""" /Record/\n arxiv_https___arxiv_org_abs_2401_04030/Similar """
GATEWAY_INTERFACE	"CGI/1.1"
SERVER_PROTOCOL	"HTTP/1.1"
REQUEST_METHOD	"GET"
QUERY_STRING	""
REQUEST_URI	"/Record/%0Aarxiv_https___arxiv_org_abs_2401_04030/Similar"
SCRIPT_NAME	"/index.php"
PHP_SELF	"/index.php"
REQUEST_TIME_FLOAT	1790731366.7892
REQUEST_TIME	1790731366

Environment Variables empty

Registered Handlers

0. Whoops\Handler\PrettyPageHandler

Similar Items

Similar Items

Environment & details:

Environment & details: