Permute, Graph, Map, Derange

1 Permute

For fixed $n$ , the sequences ${L_{k, n} : 1 \leq k \leq n}$ and ${S_{k, n} : 1 \leq k \leq n}$ constitute probability mass functions (upon normalization by $b_{n} = n!$ ) These have corresponding means $_{L} μ_{n}$ , $_{S} μ_{n}$

and variances

_{L} σ_{n}^{2}

_{S} σ_{n}^{2}

given in Table 1. We also provide the median

_{L} ν_{n}

; note that

_{S} ν_{n} = 1

for

n \geq 3

is trivial. For convenience (in table headings only), the following notation is used:

\begin{matrix} _{L} {˜ μ}_{n} = \frac{_{L} μ_{n}}{n}, & _{L} {˜ σ}_{n}^{2} = \frac{_{L} σ_{n}^{2}}{n^{2}}, & _{L} {˜ ν}_{n} = \frac{_{L} ν_{n}}{n}, \end{matrix}

\begin{matrix} _{S} {˜ μ}_{n} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{_{S} μ_{n}}{ln (n)} & if a = 1, \frac{_{S} μ_{n}}{n^{1 / 2}} & if a = 1 / 2; \end{matrix} & _{S} {˜ σ}_{n}^{2} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{_{S} σ_{n}^{2}}{n} & if a = 1, \frac{_{S} σ_{n}^{2}}{n^{3 / 2}} & if a = 1 / 2. \end{matrix} \end{matrix}

$n$	$_{L} {˜ μ}_{n}$	$_{L} {˜ σ}_{n}^{2}$	$_{L} {˜ ν}_{n}$	$_{S} {˜ μ}_{n}$	$_{S} {˜ σ}_{n}^{2}$
1000	0.624642	0.036945	0.6060	0.717352	1.307043
2000	0.624486	0.036926	0.6060	0.703135	1.307125
3000	0.624434	0.036920	0.6063	0.695960	1.307153
4000	0.624408	0.036917	0.6062	0.691295	1.307167

Table 1: Statistics for Permute ( $a = 1$ )

We have

lim n \to \infty \frac{_{L} μ_{n}}{n} =_{L} G_{1} (1, 1) = 0.624329988 54355087099...,

lim n \to \infty \frac{_{L} σ_{n}^{2}}{n^{2}} =_{L} G_{1} (1, 2) -_{L} G_{1} (1, 1)^{2} = 0.03690783006485220217...,

lim n \to \infty \frac{_{L} ν_{n}}{n} = \frac{1}{\sqrt{e}} = 0.6065306 5971263342360...,

lim n \to \infty \frac{_{S} μ_{n}}{ln (n)} = e^{- γ} = 0.561459483 56688516982...,

lim n \to \infty \frac{_{S} σ_{n}^{2}}{n} =_{S} G_{1} (1, 2) = 1.30 720779891056809974....

It is not surprising that $_{S} σ_{n}^{2}$ enjoys linear growth: $S_{1, n} \sim (1 - 1 / e) n!$ and $S_{n, n} = (n - 1)!$ jointly place considerable weight on the distributional extremes. The unusual logarithmic growth of $_{S} μ_{n}$ is due to $S_{1, n}$ nevertheless overwhelming all other $S_{k, n}$ .

A one-line proof of the $_{L} ν_{n}$ result is [10, 11]

lim n \to \infty n \sum k = ⌊ \frac{n}{\sqrt{e}} ⌋ \frac{1}{k} = lim n \to \infty ln (n) - ln (\frac{n}{\sqrt{e}}) = ln (\frac{1}{\sqrt{e}}) = \frac{1}{2} .

Alternatively, the asymptotic probability that the longest cycle has size $> n x$ is [1]

1 \int x \frac{1}{y} d y = \frac{1}{2}

hence $ln (1 / x) = 1 / 2$ and $x = 1 / \sqrt{e}$ . We will see a variation of this approach later.

No formula for the covariance between sizes of the longest cycle and shortest cycle is known. Interplay between the number of cycles and either of the extremes likewise remains inscrutable. We earlier examined not permutations, but instead integer compositions, finding a complicated recursion for a certain bivariate probability distribution

[12, 13]

. Thus a cross-correlation can be estimated for compositions, but not yet for permutations

[14].

2 Graph

Let us explain why $L_{3, 6} = 10$ , $L_{6, 6} = 60$ ; $L_{4, 7} = 105$ , $L_{7, 7} = 360$ ; and $L_{4, 8} = 315$ , $L_{5, 8} = 672$ , $L_{8, 8} = 2520$ . When $n = 6$ , a $2$ -regular graph is either a hexagon, with $5! / 2$ distinct labelings, or the disjoint union of two triangles, with $(\frac{6}{3}) / 2$ labelings. When $n = 7$ , a $2$ -regular graph is either a heptagon, with $6! / 2$ distinct labelings, or the disjoint union of a triangle and a square, with $3 (\frac{7}{3})$ labelings (since $3! / 2 = 3$ ). When $n = 8$ , a $2$ -regular graph is either an octagon, with $7! / 2$ distinct labelings; the disjoint union of a triangle and a pentagon, with $12 (\frac{7}{3})$ labelings (since $4! / 2 = 12$ ); or the disjoint union of two squares, with $9 (\frac{8}{4}) / 2$ labelings (since $(3! / 2)^{2} = 9$ ). Circumstances become more complicated when $n = 9$ : a $2$ -regular graph is either an enneagon, or the disjoint union of a square and a pentagon, or the disjoint union of three triangles, or the disjoint union of a triangle and a hexagon [15].

Upon normalization by $b_{n}$ , we obtain

$n$	$_{L} {˜ μ}_{n}$	$_{L} {˜ σ}_{n}^{2}$	$_{L} {˜ ν}_{n}$	$_{S} {˜ μ}_{n}$	$_{S} {˜ σ}_{n}^{2}$
1000	0.758771	0.037099	0.7860	3.007677	2.097084
2000	0.758297	0.037053	0.7865	3.029960	2.096470
3000	0.758139	0.037038	0.7863	3.039930	2.096262
4000	0.758060	0.037030	0.7865	3.045902	2.096157

Table 2: Statistics for Graph ( $a = 1 / 2$ )

and

lim n \to \infty \frac{_{L} μ_{n}}{n} =_{L} G_{1 / 2} (1, 1) = 0.7578230 1126849283774...,

lim n \to \infty \frac{_{L} σ_{n}^{2}}{n^{2}} =_{L} G_{1 / 2} (1, 2) -_{L} G_{1 / 2} (1, 1)^{2} = 0.03700721658229030320...,

lim n \to \infty \frac{_{L} ν_{n}}{n} = \frac{4 e}{(1 + e)^{2}} = 0.78644 773296592741014...,

lim n \to \infty \frac{_{S} μ_{n}}{n^{1 / 2}} = e^{3 / 4}_{S} G_{1 / 2} (1, 1) = 3.08504247563149222958...,

lim n \to \infty \frac{_{S} σ_{n}^{2}}{n^{3 / 2}} = e^{3 / 4}_{S} G_{1 / 2} (1, 2) = 2.09583743942571712967....

Some non-explicit formulas for the latter two results arise in Section 7. A proof for the median result is deferred to Section 3.

3 Map

Let us explain why $S_{1, 2} = 1$ , $S_{2, 2} = 3$ and $S_{1, 3} = 10$ , $S_{3, 3} = 17$ . The unique $2$ -map with totally disconnected nodes is the identity map; we associate this map with its image sequence $12$ , i.e., two isolated loops (two components of size $1$ ). The maps $11$ and $22$ are each pictured as one loop attached to a $1$ -tail (a component of size $2$ ); the map $21$ is pictured as a $2$ -cycle (again, a component of size $2$ ). For $3$ -maps, we have

\begin{matrix} 123 & i.e., three isolated loops; \end{matrix}

\begin{matrix} 113, 121, 122, 133, 223, 323 & i.e., one isolated loop % and one loop attached to a 1 -tail; \end{matrix}

\begin{matrix} 213, 321, 132 & i.e., one isolated loop and one 2 -cycle \end{matrix}

which give ten cases, and

\begin{matrix} 112, 131, 221, 322, 233, 313 & i.e., one loop attached to% a 2 -tail; \end{matrix}

\begin{matrix} 111, 222, 333 & i.e., one loop attached to two 1 % - t a i l s; \end{matrix}

\begin{matrix} 211, 212, 232, 311, 331, 332 & i.e., one 2 -cycle % attached to a 1 -tail; \end{matrix}

\begin{matrix} 231, 312 & i.e., one 3 -cycle \end{matrix}

which give seventeen cases [16, 17]. The complexity grows when $n = 4$ : it can be shown that $S_{1, 4} = 87$ , $S_{2, 4} = 27$ , $S_{4, 4} = 142$ .

Upon normalization by $b_{n} = n^{n}$ , we obtain

$n$	$_{L} {˜ μ}_{n}$	$_{L} {˜ σ}_{n}^{2}$	$_{L} {˜ ν}_{n}$	$_{S} {˜ μ}_{n}$	$_{S} {˜ σ}_{n}^{2}$
1000	0.762505	0.036968	0.7920	1.969526	1.384968
2000	0.761122	0.036980	0.7905	1.991932	1.389355
3000	0.760512	0.036985	0.7900	2.002505	1.391309
4000	0.760149	0.036988	0.7895	2.009048	1.392477

Table 3: Statistics for Map ( $a = 1 / 2$ )

and

lim n \to \infty \frac{_{L} μ_{n}}{n} =_{L} G_{1 / 2} (1, 1) = 0.7578230 1126849283774...,

lim n \to \infty \frac{_{L} σ_{n}^{2}}{n^{2}} =_{L} G_{1 / 2} (1, 2) -_{L} G_{1 / 2} (1, 1)^{2} = 0.03700721658229030320...,

lim n \to \infty \frac{_{L} ν_{n}}{n} = \frac{4 e}{(1 + e)^{2}} = 0.78644 773296592741014...,

lim n \to \infty \frac{_{S} μ_{n}}{n^{1 / 2}} = \sqrt{2}_{S} G_{1 / 2} (1, 1) = 2.06089224152016653900...,

lim n \to \infty \frac{_{S} σ_{n}^{2}}{n^{3 / 2}} = \sqrt{2}_{S} G_{1 / 2} (1, 2) = 1.40007638550124502818....

As before, some non-explicit formulas for the latter two results arise in Section 7. The asymptotic probability that the largest component has size $> n x$ is [18, 19]

1 \int x \frac{1}{2 y \sqrt{1 - y}} d y = \frac{1}{2}

hence

\frac{1}{2} ln (\frac{1 + \sqrt{1 - x}}{1 - \sqrt{1 - x}}) = \frac{1}{2}

and $x = 4 e / (1 + e)^{2}$ .

4 Derange

Derangements are permutations with no fixed points [20]. It is easy to show that $L_{3, 5} = 20$ (a longest cycle in a $5$ -derangement cannot have size $2$ or $4$ ) and $S_{2, 5} = 20$ . Upon normalization by $b_{n}$ , we obtain

$n$	$_{L} {˜ μ}_{n}$	$_{L} {˜ σ}_{n}^{2}$	$_{L} {˜ ν}_{n}$	$_{S} {˜ μ}_{n}$	$_{S} {˜ σ}_{n}^{2}$
1000	0.625266	0.037018	0.6060	1.701217	3.551193
2000	0.624798	0.036963	0.6065	1.685257	3.552276
3000	0.624642	0.036945	0.6067	1.677202	3.552637
4000	0.624564	0.036935	0.6065	1.671965	3.552818

Table 4: Statistics for Derange ( $a = 1$ )

and

lim n \to \infty \frac{_{L} μ_{n}}{n} =_{L} G_{1} (1, 1) = 0.624329988 54355087099...,

lim n \to \infty \frac{_{L} σ_{n}^{2}}{n^{2}} =_{L} G_{1} (1, 2) -_{L} G_{1} (1, 1)^{2} = 0.03690783006485220217...,

lim n \to \infty \frac{_{L} ν_{n}}{n} = \frac{1}{\sqrt{e}} = 0.6065306 5971263342360...,

lim n \to \infty \frac{_{S} μ_{n}}{ln (n)} = e^{1 - γ} = 1.52620511 159586388047...,

lim n \to \infty \frac{_{S} σ_{n}^{2}}{n} = e \cdot_{S} G_{1} (1, 2) = 3.55335920579854297440....

The asymptotic expression for the average shortest cycle length follows from

\frac{_{S} μ_{n} \cdot b_{n} + 1 \cdot (n! - b_{n})}{n!} \sim e^{- γ} ln (n)

and the fact that $b_{n} / n! \to 1 / e$ as $n \to \infty$ ; similarly for higher moments.

5 Generalized Dickman Rho (I)

Define $b_{n, m}$ to be the number of $n$ -objects whose largest component has size $\leq m$ ; thus $b_{n} = b_{n, n}$ . Given $a > 0$ , let

ρ_{a} (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 1 & if 0 \leq x < 1, 1 - a x \int 1 \frac{ρ_{a} (t - 1) \cdot (t - 1)^{a - 1}}{t^{a}} d t & if x \geq 1 \end{matrix}

and observe that the standard Dickman function $ρ (x) = ρ_{1} (x)$ . A theorem proved in [21] asserts that

for any $x > 1$ . Of course,

m \sum j = 1 L_{j, n} = b_{n, m}

hence we can easily verify this result experimentally.

$m ∖ x$	2	3	4	5	2	3	4
100	0.309347	0.049634	0.0050952	0.0003748	0.117715	0.0082644	0.0003680
200	0.308101	0.049121	0.0050026	0.0003646	0.118178	0.0082399	0.0003638
300	0.307685	0.048950	0.0049719	0.0003613	0.118329	0.0082309	0.0003624
400	0.307477	0.048864	0.0049566	0.0003597	0.118404	0.0082262	0.0003616
500	0.307353	0.048813	0.0049475	0.0003587	0.118449	0.0082233	0.0003612
600	0.307269	0.048779	0.0049414	0.0003580	0.118478	0.0082214	0.0003609
700	0.307210	0.048755	0.0049370	0.0003575	0.118500	0.0082200	0.0003607
800	0.307165	0.048736	0.0049337	0.0003572	0.118516	0.0082190	0.0003605
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$\infty$	0.306853	0.048608	0.0049109	0.0003547	0.118626	0.0082115	0.0003594

Table 5A: Ratio $b_{⌊ x m ⌋, m} / b_{⌊ x m ⌋}$ for Permute ( $a = 1$ ) and Graph ( $a = 1 / 2$ )

$m ∖ x$	2	3	4	2	3	4	5
100	0.111305	0.0074576	0.0003185	0.304359	0.048597	0.0049699	0.0003645
200	0.112756	0.0076060	0.0003258	0.305604	0.048605	0.0049409	0.0003596
300	0.113579	0.0076901	0.0003302	0.306020	0.048607	0.0049310	0.0003580
400	0.114124	0.0077458	0.0003332	0.306228	0.048608	0.0049260	0.0003572
500	0.114518	0.0077862	0.0003354	0.306353	0.048608	0.0049230	0.0003567
600	0.114822	0.0078173	0.0003371	0.306436	0.048608	0.0049210	0.0003564
700	0.115065	0.0078423	0.0003384	0.306496	0.048608	0.0049196	0.0003561
800	0.115265	0.0078629	0.0003396	0.306540	0.048608	0.0049185	0.0003559
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$\infty$	0.118626	0.0082115	0.0003594	0.306853	0.048608	0.0049109	0.0003547

Table 5B: Ratio $b_{⌊ x m ⌋, m} / b_{⌊ x m ⌋}$ for Map ( $a = 1 / 2$ ) and Derange ( $a = 1$ )

Why have we devoted effort to evaluating Dickman’s rho? Answer: the function $ρ_{a} (x)$ is fundamentally connected to $L_{k, n}$ asymptotics in Sections 1–4. The $h^{th}$ moments of the largest component size are

\begin{matrix} \infty \int 0 \frac{ρ_{1} (x)}{(x + 1)^{h + 1}} d x & and & \frac{1}{2} \infty \int 0 \frac{ρ_{1 / 2} (x)}{\sqrt{x} (x + 1)^{h + 1 / 2}} d x, & h = 1, 2, \dots \end{matrix}

for $a = 1$ and $a = 1 / 2$ , respectively. Extension to arbitrary $a > 0$ is possible. Of course, we also have integrals $_{L} G_{a} (r, h)$ available.

6 Generalized Buchstab Omega (I)

Define $b_{n, m}$ to be the number of $n$ -objects whose smallest component has size $\geq m$ ; note that $c_{n} = b_{n, n}$ . Given $a > 0$ , let

Ω_{a} (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 1 & if 1 \leq x < 2, 1 + a x \int 2 \frac{Ω_{a} (t - 1)}{t - 1} d t & if % x \geq 2 \end{matrix}

and observe that the standard Buchstab function $ω (x) = Ω_{1} (x) / x$ . A theorem proved in [22] asserts that

for any $x > 1$ . Of course,

n \sum j = m S_{j, n} = b_{n, m}

hence we can easily verify this result experimentally.

As an aside, $c_{n} / b_{n, m}$ is called the probability of connectedness in [22]

, i.e., the odds that an

n

-object, whose smallest component has size at least

m

, is connected. No analogous name has been proposed for

b_{n, m} / b_{n}

from Section 5, i.e., the odds that all components of an

n

-object have size at most

m

. Maybe probability of smoothness would be appropriate (“smooth” coming from prime number theory). For Section 7, the same ratio might be called the probability of roughness, wherein all components of an

n

-object have size at least

m

$m ∖ x$	2	3	4	5	2	3	4	5
100	0.990	0.587992	0.443034	0.354438	0.995	0.740555	0.628689	0.555092
200	0.995	0.589306	0.444151	0.355327	0.997	0.741591	0.629581	0.555860
300	0.997	0.589743	0.444523	0.355624	0.998	0.741936	0.629878	0.556116
400	0.997	0.589962	0.444710	0.355772	0.999	0.742108	0.630027	0.556244
500	0.998	0.590093	0.444822	0.355862	0.999	0.742212	0.630116	0.556321
600	0.998	0.590180	0.444896	0.355921	0.999	0.742281	0.630175	0.556372
700	0.999	0.590242	0.444949	0.355963	0.999	0.742330	0.630218	0.556409
800	0.999	0.590289	0.444989	0.355995	0.999	0.742367	0.630250	0.556436
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$\infty$	1	0.590616	0.445269	0.356218	1	0.742626	0.630473	0.556628

Table 6A: Ratio $c_{⌊ x m ⌋} / b_{⌊ x m ⌋, m}$ for Permute ( $a = 1$ ) and Graph ( $a = 1 / 2$ )

$m ∖ x$	2	3	4	5	2	3	4	5
100	0.995	0.746112	0.635215	0.561960	0.990	0.587992	0.443034	0.354438
200	0.998	0.745502	0.634175	0.560698	0.995	0.589306	0.444151	0.355327
300	0.998	0.745124	0.633623	0.560060	0.997	0.589743	0.444523	0.355624
400	0.999	0.744866	0.633267	0.559656	0.997	0.589962	0.444710	0.355772
500	0.999	0.744677	0.633013	0.559371	0.998	0.590093	0.444822	0.355862
600	0.999	0.744530	0.632819	0.559156	0.998	0.590180	0.444896	0.355921
700	0.999	0.744412	0.632664	0.558985	0.999	0.590242	0.444949	0.355963
800	0.999	0.744314	0.632537	0.558845	0.999	0.590289	0.444989	0.355995
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$\infty$	1	0.742626	0.630473	0.556628	1	0.590616	0.445269	0.356218

Table 6B: Ratio $c_{⌊ x m ⌋} / b_{⌊ x m ⌋, m}$ for Map ( $a = 1 / 2$ ) and Derange ( $a = 1$ )

Buchstab’s Omega, as defined here, does not seem to be allied with $S_{k, n}$ asymptotics in Sections 1–4. A different generalization is discussed in Section 7.

7 Generalized Buchstab Omega (II)

Define $b_{n, m}$ to be the number of $n$ -objects whose smallest component has size $\geq m$ (as in Section 6). When restricting attention to permutations, Panario & Richmond [3] obtained that

lim m \to \infty \frac{m b_{⌊ x m ⌋, m}}{b_{⌊ x m ⌋}} = ω (x)

for any $x > 1$ , where $ω (x)$ is the standard Buchstab function. They seemed to presume that the same limit would occur for derangements (since both permutations and derangements have $a = 1$ ), which is not true. Replace now the initial factor $m$ in the numerator by $m^{1 / 2}$ . Panario & Richmond realized that $2$ -regular graphs and mappings would possess a limit different from $ω (x)$ . They seemed, however, to presume that equivalent limits would occur (since both graphs and maps have $a = 1 / 2$ ), which is again untrue. In Section 5, we studied two functions $Ω_{a}$ , $a \in {1, 1 / 2}$ ; here we have four omega (lowercase “o”) functions $ω_{A}$ , $A \in {P, G, M, D}$ , one for each structure under consideration. Upon multiplication of limits, we discover

\frac{ω_{A} (x)}{Ω_{a} (x)} = lim m \to \infty \frac{m^{a} c_{⌊ x m ⌋}}{b_{⌊ x m ⌋}} = \frac{1}{x^{a}} \cdot ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 1 & if A = P and a = 1, e^{3 / 4} \sqrt{π} / 2 & if A = G and a = 1 / 2, \sqrt{π / 2} & if A = M and a = 1 / 2, e & if A = D and a = 1 \end{matrix}

using known $c_{n} / b_{n}$ asymptotics as $n \to \infty$ for graphs and mappings [23, 24, 25]. Perhaps, for fixed $a > 0$ , $ω_{A}$ varies only up to multiplicative constant. These formulas allow us to provide numerical values in the final rows of Tables 6A and 6B.

Return now to Panario & Richmond. Especially puzzling is a claim (for permutations) that [3]

lim n \to \infty \frac{_{S} σ_{n}^{2}}{n} = \infty \int 2 \frac{ω (x)}{x^{2}} d x .

From Section 1, the left-hand side is $_{S} G_{1} (1, 2) = 1.307207...$ [4] whereas the right-hand side is $\frac{1}{2} (0.556816...)$ [26]. Thus predictions in [3] for $A \in {P, D}$ are evidently mistaken.

$m ∖ x$	2	3	4	5	2	3	4	5
100	0.50500	0.56690	0.56429	0.56427	1.33744	1.46573	1.49445	1.51342
200	0.50250	0.56564	0.56287	0.56286	1.33203	1.46216	1.49116	1.51038
300	0.50166	0.56522	0.56240	0.56239	1.33023	1.46097	1.49007	1.50937
400	0.50125	0.56501	0.56216	0.56216	1.32933	1.46037	1.48952	1.50887
500	0.50100	0.56488	0.56202	0.56202	1.32879	1.46002	1.48920	1.50856
600	0.50083	0.56480	0.56193	0.56192	1.32843	1.45978	1.48898	1.50836
700	0.50072	0.56474	0.56186	0.56186	1.32817	1.45961	1.48882	1.50822
800	0.50063	0.56470	0.56181	0.56181	1.32798	1.45948	1.48871	1.50811
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$\infty$	0.5	0.56438	0.56146	0.56145	1.32663	1.45860	1.48789	1.50735

Table 7A: Ratio $m^{a} b_{⌊ x m ⌋, m} / b_{⌊ x m ⌋}$ for Permute ( $a = 1$ ) and Graph ( $a = 1 / 2$ )

$m ∖ x$	2	3	4	5	2	3	4	5
100	0.87413	0.95520	0.97361	0.98570	1.37273	1.54100	1.53390	1.53385
200	0.87676	0.96022	0.97895	0.99132	1.36594	1.53756	1.53005	1.53001
300	0.87816	0.96259	0.98148	0.99397	1.36367	1.53642	1.52876	1.52874
400	0.87906	0.96406	0.98303	0.99559	1.36254	1.53585	1.52812	1.52810
500	0.87971	0.96508	0.98411	0.99672	1.36186	1.53551	1.52774	1.52772
600	0.88021	0.96584	0.98492	0.99756	1.36141	1.53528	1.52748	1.52746
700	0.88060	0.96644	0.98555	0.99823	1.36109	1.53512	1.52730	1.52728
800	0.88092	0.96693	0.98607	0.99876	1.36084	1.53500	1.52716	1.52715
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$\infty$	0.88623	0.97438	0.99395	1.00695	1.35914	1.53415	1.52620	1.52619

Table 7B: Ratio $m^{a} b_{⌊ x m ⌋, m} / b_{⌊ x m ⌋}$ for Map ( $a = 1 / 2$ ) and Derange ( $a = 1$ )

Why have we devoted effort to evaluating Buchstab’s omega? Answer: an array of formulas, parallel to those involving $ρ_{a} (x)$ , corresponding to $h^{th}$ moments of the smallest component size, were proposed in [3]:

\begin{matrix} \infty \int 2 \frac{ω_{A} (x)}{x^{h + 1 / 2}} d x, & h = 1, 2, \dots, & for A \in {G, M} \end{matrix}

and would be exceedingly attractive. Unfortunately the potential for fulfillment is not good. No high-precision numerical estimates of these integrals are currently known; thus we are not certain that any of the various $ω_{A} (x)$ are necessarily allied with $S_{k, n}$ asymptotics in Sections 1–4. For now, the formulas remain frustratingly non-explicit and unverified.

8 Generalized Dickman Rho (II)

Define $b_{n, m}$ to be the number of $n$ -objects whose largest component has size $\leq m$ (as in Section 5). When restricting attention to permutations, we observe that

lim m \to \infty \frac{⌊ x m ⌋ c_{⌊ x m ⌋}}{b_{⌊ x m ⌋, m}} = \frac{1}{ρ (x)}

for any $x > 1$ , where $ρ (x)$ is the standard Dickman function. When restricting attention to derangements, a factor of $e$ needs to be included (just as in Sections 4 and 7). Replace now the initial factor $⌊ x m ⌋$ in the numerator by ${⌊ x m ⌋}^{1 / 2}$ . Again $2$ -regular graphs and mappings possess non-equivalent limits different from $1 / ρ (x)$ . Just as we found the defining limit for Buchstab’s omega in terms of $Ω_{a} (x)$ earlier, here we discover the limit in terms of $ρ_{a} (x)$ :

We know that $ρ_{a}$ is important (Section 5) and believe that $ω_{A}$ deserves further study (Section 7). It is hoped that someone else might succeed in carrying on research where we have stopped.

$m ∖ x$	2	3	4	5	2	3	4	5
100	3.23262	20.1473	196.264	2668.39	15.9879	227.490	5106.02	161434.
200	3.24569	20.3580	199.898	2742.41	15.9005	227.928	5161.08	164830.
300	3.25007	20.4291	201.130	2767.66	15.8719	228.098	5180.23	166005.
400	3.25227	20.4648	201.750	2780.41	15.8577	228.188	5189.95	166600.
500	3.25359	20.4863	202.124	2788.09	15.8492	228.244	5195.83	166961.
600	3.25447	20.5006	202.373	2793.22	15.8436	228.281	5199.78	167202.
700	3.25510	20.5109	202.552	2796.90	15.8396	228.308	5202.60	167375.
800	3.25558	20.5186	202.686	2799.66	15.8365	228.329	5204.73	167504.
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$\infty$	3.25889	20.5726	203.628	2819.09	15.8156	228.476	5219.73	168421.

Table 8A: Ratio for Permute ( $a = 1$ ) and Graph ( $a = 1 / 2$ )

$m ∖ x$	2	3	4	5	2	3	4	5
100	11.0531	165.524	3883.12	128134.	8.93117	55.9354	546.943	7458.53
200	10.9698	163.012	3811.16	125549.	8.89477	55.9254	550.162	7558.92
300	10.9164	161.547	3766.89	123839.	8.88269	55.9236	551.265	7593.21
400	10.8799	160.574	3737.04	122663.	8.87665	55.9229	551.822	7610.52
500	10.8531	159.869	3715.21	121794.	8.87304	55.9226	552.159	7620.95
600	10.8322	159.327	3698.37	121120.	8.87063	55.9224	552.384	7627.93
700	10.8155	158.894	3684.85	120577.	8.86890	55.9223	552.545	7632.92
800	10.8016	158.537	3673.69	120127.	8.86761	55.9223	552.665	7636.67
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$\infty$	10.5652	152.628	3486.92	112510.	8.85859	55.9221	553.517	7663.07

Table 8B: Ratio for Map ( $a = 1 / 2$ ) and Derange ( $a = 1$ )

9 Addendum

At the conclusion of Section 5, we gave expressions for the $h^{th}$ moments of largest component size, given $a = 1$ or $a = 1 / 2$ , without justification. Here is a plausibility argument. Assuming (absent any proof) that the first moment for arbitrary $a > 0$ is [27]

λ_{a} = 1 - \infty \int 1 \frac{ρ_{a} (x)}{x^{2}} d x,

we reverse integration-by-parts:

\begin{matrix} d u = x^{- 2 - a} d x, & v = x^{a} ρ_{a} (x), u = - \frac{1}{1 + a} x^{- 1 - a}, & d v = a [x^{- 1 + a} ρ_{a} (x) - (x - 1)^{- 1 + a} ρ_{a} (x - 1)] d x \end{matrix}

and obtain

- \frac{a}{1 + a} ⎡ ⎢ ⎣ \infty \int 1 \frac{ρ_{a} (x)}{} x^{2} d x - \infty \int 1 \frac{ρ_{a} (x - 1)}{(x - 1)^{1 - a} x^{1 + a}} d x ⎤ ⎥ ⎦ = {- \frac{1}{1 + a} \frac{ρ_{a} (x)}{x} ∣ ∣ ∣}_{1}^{\infty} - \infty \int 1 \frac{ρ_{a} (x)}{x^{2}} d x

i.e.,

\frac{a}{1 + a} \infty \int 0 \frac{ρ_{a} (x)}{x^{1 - a} (x + 1)^{1 + a}} d x = \frac{1}{1 + a} - (1 - λ_{a}) + \frac{a}{1 + a} (1 - λ_{a}) = \frac{λ_{a}}{1 + a} .

From here, we infer (again, absent any proof) that the $h^{th}$ moment is

\begin{matrix} a \infty \int 0 \frac{ρ_{a} (x)}{x^{1 - a} (x + 1)^{h + a}} d x, & h = 1, 2, \dots . \end{matrix}

An explanation for $_{S} μ_{n}$ and $_{S} σ_{n}^{2}$ asymptotics in Sections 2 and 3 was not given. Reason: it is more complicated than the proof in Section 4. At some later point, we hope to study combinatorial objects called cyclations [28] for which moments are known to be precisely $_{S} G_{1 / 2}$ . Since $\sqrt{n} c_{n} / b_{n} \to \sqrt{π} / 2$ as $n \to \infty$ for these, and because corresponding limits for graphs and mappings appear in Section 7, the factors $e^{3 / 4}$ and $\sqrt{2}$ emerge.

10 Acknowledgements

I am grateful to Nicholas Pippenger [28] for a helpful discussion. The creators of Mathematica, as well as administrators of the MIT Engaging Cluster, earn my gratitude every day. A sequel to this paper is now available [29]; another will be released soon [30].

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Steven Finch

MIT Sloan School of Management

Cambridge, MA, USA

steven_finch@harvard.edu

Permute, Graph, Map, Derange

Second Best, Third Worst, Fourth in Line

Rounds, Color, Parity, Squares

Components and Cycles of Random Mappings

Approximating the Median under the Ulam Metric

Oblivious Median Slope Selection

Ranking Median Regression: Learning to Order through Local Consensus

1 Permute

2 Graph

3 Map

4 Derange

5 Generalized Dickman Rho (I)

6 Generalized Buchstab Omega (I)

7 Generalized Buchstab Omega (II)

8 Generalized Dickman Rho (II)

9 Addendum

10 Acknowledgements

References

Permute, Graph, Map, Derange

Related Research

Second Best, Third Worst, Fourth in Line

Rounds, Color, Parity, Squares

Components and Cycles of Random Mappings

Approximating the Median under the Ulam Metric

Oblivious Median Slope Selection

Ranking Median Regression: Learning to Order through Local Consensus

1 Permute

2 Graph

3 Map

4 Derange

5 Generalized Dickman Rho (I)

6 Generalized Buchstab Omega (I)

7 Generalized Buchstab Omega (II)

8 Generalized Dickman Rho (II)

9 Addendum

10 Acknowledgements

References