Probability of Indecomposability of a Random Mapping Function

Katz, Leo

doi:10.1214/aoms/1177728496

Cited by 90 publications

(45 citation statements)

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“…Similar results have been found for some restricted classes of unary functions, for example random " hollow mappings", i.e. functions without fixed points, in [13] and random permutations in L. Shepp and S. Lloyd [21].…”

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confidence: 80%

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Probabilities of First-Order Sentences about Unary Functions

Lynch¹

1985

Transactions of the American Mathematical Society

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Abstract.Let / be any fixed positive integer and a a sentence in the first-order predicate calculus of/unary functions. For positive integers n, an «-structure is a model with universe {0,1,...,« -1} and/unary functions, and p(n,a) is the ratio of the number of «-structures satisfying a to n"f, the number of «-structures. We show that lim"_00/i(«, a) exists for all such a, and its value is given by an expression consisting of integer constants and the operators +,-,,/, and ex.

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confidence: 80%

“…M. Kruskal [14] subsequently proved that it was \ogen + C + o(l), where C = 0.5772... is Euler's constant. L. Katz [13] found an expression for the probability that a random function is connected (i.e. has only one component), and showed that it was asymptotic to (tr/2n)1/2.…”

mentioning

confidence: 99%

Probabilities of First-Order Sentences about Unary Functions

Lynch¹

1985

Transactions of the American Mathematical Society

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“…This follows easily from the tree volume formula by iteration of the argument leading to (7). L The instance of (31), with x s -1 for all s ¥ W and x 0 a positive integer, was discovered by Katz [25], who used it to show that the number C(n, k) of mappings from an n element set to itself whose digraph is connected with exactly k cyclic points is…”

Section: Hurwitz Identitiesmentioning

confidence: 99%

Forest Volume Decompositions and Abel–Cayley–Hurwitz Multinomial Expansions

Pitman

2002

Journal of Combinatorial Theory, Series A

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This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes defined by the enumerator polynomials of sets of rooted labeled forests. These decompositions involve the following basic forest volume formula, which is a refinement of Cayley's multinomial expansion: for R ı S the polynomial enumerating out-degrees of vertices of rooted forests labeled by S whose set of roots is R, with edges directed away from the roots, is

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“…Next, the number of connected graphs with n labeled points is given in [2] by «-i n k Hence W n = n(C n~Rn ).…”

Section: Hencementioning

confidence: 99%

The enumeration of rooted trees by total height

Riordan

Sloane

1969

J. Aust. Math. Soc.

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The height (as in [3] and [4]) of a point in a rooted tree is the length of the path (that is, the number of lines in the path) from it to the root; the total height of a rooted tree is the sum of the heights of its points. The latter arises naturally in studies of random neural networks made by one of us (N.J.A.S.), where the enumeration of greatest interest is that of trees with all points distinctly labeled.

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Probability of Indecomposability of a Random Mapping Function

Cited by 90 publications

References 1 publication

Probabilities of First-Order Sentences about Unary Functions

Probabilities of First-Order Sentences about Unary Functions

Forest Volume Decompositions and Abel–Cayley–Hurwitz Multinomial Expansions

The enumeration of rooted trees by total height

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