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Topic: Stirling numbers of the second kind

	PlanetMath: Stirling numbers of the second kind
		are a doubly indexed sequence of natural numbers, enjoying a wealth of interesting combinatorial properties.
		Indeed, the Stirling numbers of the second kind can be characterized as the coefficients involved in the corresponding change of basis matrix, i.e.
		This is version 8 of Stirling numbers of the second kind, born on 2002-03-29, modified 2007-03-28.
	planetmath.org /encyclopedia/StirlingNumbersSecondKind.html (558 words)

	Stirling number - Wikipedia, the free encyclopedia
		Stirling numbers of the first kind are written with a small s, and those of the second kind with a large S (Abramowitz and Stegun use an uppercase S and a flletter S respectively).
		The absolute value of the Stirling number of the first kind, s(n,k), counts the number of permutations of n objects with exactly k orbits (equivalently, with exactly k cycles).
		Stirling numbers of the second kind S(n,k) (with a capital "S") count the number of equivalence relations having k equivalence classes defined on a set with n elements.
	en.wikipedia.org /wiki/Stirling_number (1368 words)

	PlanetMath: Stirling numbers of the first kind
		To be more precise, the defining relation for the Stirling numbers of the first kind is:
		Q.E.D. "Stirling numbers of the first kind" is owned by rmilson.
		This is version 3 of Stirling numbers of the first kind, born on 2002-03-31, modified 2004-11-01.
	planetmath.org /encyclopedia/StirlingNumbers.html (368 words)

	Stirling biography
		Tweddle [James Stirling: this about series and such things (Edinburgh, 1988).',3)" onmouseover="window.status='Click to see reference';return true">3] notes that a student with the name 'James Stirling' matriculated at the University of Edinburgh on 24 March 1710, did not graduate, and has a signature which is similar to that of the mathematician.
		Stirling seems to have been promised a chair of mathematics in Venice but, for some reason that is not known, the appointment fell through.
		Newton proposed Stirling for a fellowship of the
	www-history.mcs.st-andrews.ac.uk /history/Biographies/Stirling.html (2698 words)

	Math Forum - Ask Dr. Math
		Suppose we let A be the number of ways that cell(1) is empty, B is the number of ways that cell(2) is empty and C is the number of ways that cell (3) is empty.
		This is the number of ways of distributing n distinct objects into m identical cells such that no cell is empty.
		Stirling Numbers of the First Kind ----------------------------------- In general S1(n,m) is the number of ways to partition n objects into m (non-empty) parts and arrange the members of each part around a circle.
	mathforum.org /library/drmath/view/51550.html (1190 words)

	Dictionary of Combinatorics -- S (Site not responding. Last check: 2007-10-20)
		The first type of complex can be made into the second, by embedding the members of the sets in general position in some space of suffiently high dimension, and forming simplices by taking the convex hull of each set.
		The Stirling numbers of the second kind, S(n,t), have the combinatorial interpretation of counting the number of ways of partitioning an n element set into t non-empty subsets.
		The Stirling numbers of the first kind, s(n,t), are the coefficients on the monomials in an expansion of polynomials of the form
	www.southernct.edu /~fields/comb_dic/S.html (533 words)

	Essays/Stirling Numbers - J Wiki
		Stirling numbers of the first kind and Stirling numbers of the second kind can be computed as follows:
		The Stirling numbers of the first and second kinds are related in a simple way:
		That is, the table of Stirling numbers of one kind are the absolute values of the matrix inverse of the table of Stirling numbers of the other kind.
	www.jsoftware.com /jwiki/Essays/Stirling_Numbers (157 words)

	Stirling's Triangle for Subsets
		The Stirling numbers of the second kind are denoted
		Stirling numbers of the second kind obey the recurrence relations
		Dickau, R. ``Stirling Numbers of the Second Kind.'' http://forum.swarthmore.edu/advanced/robertd/stirling2.html.
	steiner.math.nthu.edu.tw /chuan/123/test/stirling-subsets.htm (199 words)

	GAP Manual: 46.5. Stirling2 (Site not responding. Last check: 2007-10-20)
		Stirling numbers of the second kind are defined by S_2(0,0) = 1, S_2(n,0) = S_2(0,k) = 0 if n, k <> 0 and the recurrence S_2(n,k) = k S_2(n-1,k) + S_2(n-1,k-1).
		Stirling numbers of the second kind appear as coefficients in the expansion of x^n = sum_(k=0)^(n)(S_2(n,k) k!
		There are many formulae relating Stirling numbers of the second kind to Stirling numbers of the first kind, Bell numbers, and Binomial numbers.
	www.komatsu-c.ac.jp /~yanagiha/GAP-Manual/Stirling2.html (105 words)

	Dynamics And Hierarchies
		An enumeration of the classification diagrams for different number of items produces the integer sequence 1, 1, 4, 236, 2752, which the EIS list as sequence A000311. The connection between set partitions and hierarchies occur in several places, the hierarchies of n are also referred to as the total partitions of n, and hierarchical partitions.
		The reason for forbidding Bell number diagram with a single partition in classification diagrams becomes clear, the terms are absent from the right hand side of the equation as they are the terms we wish to solve for.
		Only the second item added to a partition affects the number of ways that the sets partition can be expressed; because the partitions are indistinguishable, the association of the first item with its partition is arbitrary.
	www.tetration.org /Dynamics/DynamicsAndHierarchies.htm (6678 words)

	id:A008277 - OEIS Search Results (Site not responding. Last check: 2007-10-20)
		S2(3,2)=3 since the number of partitions of {1,2,3,4} into three subsets of nonconsecutive integers is 3, i.e., 1324, 1423, 1243.
		Korshunov, A. D., Asymptotic behavior of Stirling numbers of the second kind.
		E. Weisstein, Stirling numbers of the 2nd kind.
	www.research.att.com /~njas/sequences/A008277 (839 words)

	Information on Set Partitions
		The number of partitions of an n-set into k blocks is called a Stirling number of the second kind and is denoted S(n,k).
		Sometimes ther Stirling numbers of the second kind are written in a manner similar to the binomial coefficients, except that curly braces are used instead of parentheses.
		Information about James Stirling, for whom the Stirling numbers of the second kind are named.
	www.theory.csc.uvic.ca /~cos/inf/setp/SetPartitions.html (544 words)

	Algebraic Combinatorics -- Various combinatorial numbers
		These S(n,m) are called the Stirling numbers of the second kind.
		Another consequence of the definition of Stirling numbers of the second kind and of Corollary is
		There is a program for computing the recontre numbers and their generalization.
	www.mathe2.uni-bayreuth.de /frib/html/book/hyl00_55.html (480 words)

	Stirling Numbers of the Second Kind
		The Stirling numbers of the second kind describe the number of ways a set with n elements can be partitioned into k disjoint, non-empty subsets.
		Here are some diagrams representing the different ways the sets can be partitioned: a line connects elements in the same subset, and a point represents a singleton subset.
		The sums of the Stirling numbers of the second kind,
	www-gap.dcs.st-and.ac.uk /~history/Miscellaneous/StirlingBell/stirling2.html (117 words)

	Carlitz-Riordan Identities
		The associated Stirling numbers of the second kind were developed in J. Riordan's book, "An Introduction to Combinatorial Analysis" (Wiley, 1958).
		The second-order Eulerian numbers were introduced in the article "The Coefficients in an Asymptotic Expansion" by L. Carlitz (Proc.
		They are now denoted by <> and were shown to answer some counting problems for permutations of multisets.
	www.math.uaa.alaska.edu /~smiley/stir2eul2.html (278 words)

	GAP Manual: 46 Combinatorics
		First this package contains various functions that are related to the number of selections from a set (see Factorial, Binomial) or to the number of partitions of a set (see Bell, Stirling1, Stirling2).
		{n choose k} is the number of combinations with k elements, i.e., the number of subsets with k elements, of a set with n elements.
		The Bell numbers are defined by B(0)=1 and the recurrence B(n+1) = sum_{k=0}^{n}{{n choose k}B(k)}.
	www.mcs.kent.edu /system/documentation/gap/CHAP046.htm (2395 words)

	stirling.html (Site not responding. Last check: 2007-10-20)
		We show how computer algebra can be used to compute automatically the asymptotics of the Bell numbers and of the average value and variance of the number of parts in a partition.
		Alternatively, since these numbers count the number of partitions of a set, this generating function and the numbers can be obtained using
		The technique demonstrated above also applies to the average number of parts in a partition, where an indefinite cancellation occurs and this exemplifies the use of multiple scales.
	pauillac.inria.fr /algo/libraries/autocomb/stirling-html/stirling1.html (855 words)

	Josh Cooper's Math Pages : Whitney Numbers
		Whitney number is the number of flats in L with rank n.
		Notice how, for the trees -- which are the (connected) graphs with the fewest edges -- the numbers increase and then decrease in each row, i.e., the binomial coefficients are unimodal; and how, for the complete graphs -- which are the graphs with the most edges -- the Whitney numbers are also unimodal.
		Rota conjectured in 1970 that the sequence of Whitney numbers for every graph is unimodal, but no one has been able to prove it or find a counterexample.
	www.math.sc.edu /~cooper/graph.html (663 words)

	tetration.org - Combinatorics of Iterated Functions
		The Bell polynomials, which are related to set partitions, are the foundation of Continuous iteration of dynamical maps and the Stirling numbers of the first and second kind are also referenced.
		In Analysis of Carleman Representation of Analytical Recursions the Catalan numbers and the ballot numbers are used in deriving the continuous iteration of the logistics equation.
		I made a couple of guesses as to the sum of the coefficients and the number of terms; the OEIS indicated I was looking at combinatoric structures from Ernst Schroeder's 1870 paper Vier combinatorische Probleme.
	www.tetration.org /Combinatorics/index.html (884 words)

	Citebase - Two notes on notation
		The second notation puts Stirling numbers on the same footing as binomial coefficients.
		Since binomial coefficients are written on two lines in parentheses and read "n choose k", Stirling numbers of the first kind should be written on two lines in brackets and read "n cycle k", while Stirling numbers of the second kind should be written in braces and read "n subset k".
		The virtues of this notation are that Stirling partition numbers frequently appear in combinatorics, and that it more clearly presents functional relations similar to those satisfied by binomial coefficients.
	www.citebase.org /abstract?id=oai:arXiv.org:math/9205211 (228 words)

	Set Partitions
		As a matter of curiosity, the Bell number for a set with N members, B(N), is the sum of the number of ways that it can be partitioned into 1, 2, 3,...
		The number of partitions increases rapidly with set size - more than a billion for 15 items - so it's not practical to count the number of ways by actually generating them.
		There is a recursive definition for Stirling numbers of the second type, therefore we can sum these as described above to get the Bell number.
	www.delphiforfun.org /Programs/Math_Topics/set_partitions.htm (680 words)

	Jambands.com \| Columns::David Steinberg \| Everyone Must Add and Subtract Those 18 Steps \| 2006-09-17
		Stirling numbers of the second kind are not encounters with UFOs that will inspire you to make a model of Devil's Tower in mashed potatoes.
		Rather they are the number of ways that you can group together a set of n distinct elements into k different groups; in this case, it's how many ways you can get 3 groups out of a 4 element set.
		He named one "Stirling numbers of the first kind," and the second, "Stirling numbers of the second kind." The notation of the first kind is s(n, k) - s is for Stirling of course - and the second uses S(n, k).
	www.jambands.com /Columns/Zzyzx/content_2006_09_17.00.phtml (2997 words)

	Four Numerical Triangles in Nexus
		We deal in triangles of numbers and their 3-term-recurrences (3tr's), from which a variety of sequences are routinely extracted.
		We represent these numbers by <> and refer to their geographical positions on or under the ray m=k in the first quadrant of the (m,k) plane.
		Triangle of associated Stirling numbers of the second kind = A008299
	www.math.uaa.alaska.edu /~smiley/numbertriangle.html (1062 words)

	Gian Carlo Rota Polish Seminar
		Stirling numbers of the second kind are umbrally extended in a new way and the resulting new type of dobinskian formulae are discovered.
		Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers` triangular array are then calculated and the explicit formula for them is provided.
		A comparison with the similar extension of binomial, Gaussian and Stirling coefficients due to Konvalina [2,3] is one of the goal of the presentation.
	ii.uwb.edu.pl /akk/sem/sem_rota.htm (3088 words)

	Sequences of Polynomials
		Figure 8 shows the Stirling numbers of the second kind mod 3 organized as a triangle as well.
		Recall that Stirling numbers of the second kind are defined by
		Figure 8: Eighty rows of Stirling Numbers of the second kind mod 3
	www.cecm.sfu.ca /~loki/Papers/Numbers/node7.html (308 words)

	Pascal's Triangle - Stirling numbers
		to indicate the Stirling numbers of the first and second kinds, named for 18th Century mathematician James Stirling (1692-1770).
		Just as in Concrete Mathematics, we will talk about the second kind of Stirling numbers first because the question is a little simpler.
		Usually, the curly brackets {} refer to sets in math, so using these brackets for the Stirling numbers of the second kind is a good mnemonic.
	binomial.csuhayward.edu /stirling.html (347 words)

	Pascal's Triangle - Binomial Coefficient Identities (Site not responding. Last check: 2007-10-20)
		The second difference equation of the row sequence is zero only at these values (and their mirror images on the same row) among all rows of Pascal’s Triangle
		The authors considered this to be “the hairiest sum of binomial coefficients known”, this is the generalization of Dixon’s Identity to have more variables, both in the top entry of the binomial coefficient and in the index variables; it was conjectured by Freeman Dyson in 1962 and proved by several people shortly thereafter.
		Relation between the B.C.s and the Stirling numbers of the second kind to generate the Eulerian numbers
	www.math.uconn.edu /~troby/Hidden/4MattH/PTW/Identities.html (1556 words)

	GAP Manual: 46 Combinatorics
		Stirling numbers of the first kind are defined by S
		Stirling numbers of the second kind are defined by S
		Stirling numbers of the second kind appear as coefficients in the expansion of x
	www.institut.math.jussieu.fr /~jmichel/htm/CHAP046.htm (2277 words)

	Articles
		Cooper and R. Kennedy, "Patterns, Automata, and Stirling Numbers of the Second Kind", Mathematics and Computer Education Journal, 26 (1992), 120-124.
		Cooper and R. Kennedy, "On the Number of Occurrences of the Digit 1 in the Sequence of Positive Integers Less Than n", Journal of Institute of Mathematics and Computer Science, 7 (1994), 121-127.
		Cooper, "An Identity for Period k Second Order Linear Recurrence Systems", (accepted for publication in the Proceedings of the 12th International Conference on Fibonacci Numbers and Their Applications).
	www.math-cs.cmsu.edu /~curtisc/articles.html (1932 words)

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