In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book Methodus differentialis (1730).[1] They were rediscovered and given a combinatorial meaning by Masanobu Saka in 1782.[2]
Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second kind. Additionally, Lah numbers are sometimes referred to as Stirling numbers of the third kind. Each kind is detailed in its respective article, this one serving as a description of relations between them.
A common property of all three kinds is that they describe coefficients relating three different sequences of polynomials that frequently arise in combinatorics. Moreover, all three can be defined as the number of partitions of n elements into k non-empty subsets, where each subset is endowed with a certain kind of order (no order, cyclical, or linear).
In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in...
particularly in combinatorics, a Stirlingnumber of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects...
especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind...
Bridge and the port. Located on the River Forth, Stirling is the administrative centre for the Stirling council area, and is traditionally the county town...
the regenerator is what differentiates a Stirling engine from other closed-cycle hot air engines. In the Stirling engine, a working fluid (e.g. air) is heated...
abandoned, the S.29, which later received the name Stirling, proceeded to production. In early 1941, the Stirling entered squadron service. During its use as...
paternal grandparents were Sir William Stirling-Maxwell, 9th Baronet and Lady Anna Maria Leslie-Melville. Stirling was educated in England at the Catholic...
Peirce (1880) and Aitken (1933). Touchard polynomials Catalan numberStirlingnumberStirling numbers of the first kind Gardner 1978. Halmos, Paul R. (1974)...
Problem for n = 5 {\displaystyle n=5} , the fifth octagonal number, and the Stirlingnumber of the second kind S ( 6 , 4 ) {\displaystyle S(6,4)} that...
the programme for two series. In 2012, Stirling appeared on Russell Howard's Good News. In June 2015, Stirling became the narrator of the ITV2 reality...
Stirling Castle, located in Stirling, is one of the largest and most historically and architecturally important castles in Scotland. The castle sits atop...
University of Stirling Archives. Archived from the original on 3 September 2017. Retrieved 8 July 2017. "Art at Stirling". Stirling University. Archived...
related to Stirling Moss. Official website Stirling Moss at 24 Hours of Le Mans (in French) Grand Prix History – Hall of Fame, Stirling Moss Stirling Moss profile...
Wallace joined Moray in September near Dundee, and they marched to Stirling. Stirling, in the words of Stuart Reid, was "traditionally regarded as the key...
with RFA Stirling Castle – first of the navy's new motherships". Navy Lookout. 4 July 2023. "UK Royal Navy's Future MCM Mothership "Stirling Castle" Begins...
elements. The number of permutations of n distinct objects is n!. The number of n-permutations with k disjoint cycles is the signless Stirlingnumber of the...
singleton. The number of partitions of an n-element set into exactly k (non-empty) parts is the Stirlingnumber of the second kind S(n, k). The number of noncrossing...
Global Information