Counting Young Tableaux of Bounded Height^{∗}

François Bergeron and Francis Gascon Departement de Mathematiques Univerite du Quebec à Montreal Email addresses: bergeron.francois@uqam.ca and gascon.francis@uqam.ca

Abstract

We show that formulae of Gessel for the generating functions for Young standard tableaux of height bounded by k (see [2]) satisfy linear differential equations, with polynomial coefficients, equivalent to P - recurrences conjectured by Favreau, Krob and the first author (see [1]) for the number of bounded height tableaux and pairs of bounded height tableaux.

1. Results

Let us first fix some notation. A partition λ of a positive integer n is a sequence of integers

λ_{1} ≥ λ_{2} ≥ … ≥ λ_{k} > 0

such that ∑_{i} λ_{i} = n. We denote this by writing λ ⊢ n, and say that k is the height h(λ) of λ. The height of the empty partition (of 0) is 0. The (Ferrer’s) diagram of a partition λ is the set of points (i, j) ∈ Z^{2} such that 1 ≤ j ≤ λ_{i}. It is also denoted by λ. Clearly a partition is characterized by its diagram. The conjugate λ′ of a partition λ is the partition with diagram {(j, i) | (i, j) ∈ λ}.

A standard Young tableau T is an injective labeling of a Ferrer’s diagram by the elements of {1, 2, …, n} such that T (i, j) < T (i + 1, j) for 1 ≤ i < k and T (i, j) < T (i, j + 1) for 1 ≤ j < λ_{i}. We further say that λ is the shape of the tableau T. For a given λ, the number f_{λ} of tableaux of shape λ is given by the hook length formula

f_{λ} = \frac{n!}{∏_{c} h_{c}} ,

where c = (i, j) runs over the set of points in the diagram of λ, and

h_{c} = λ_{i} − i + λ′_{j} − j + 1.

Other classical results in this context are

∑_{λ⊢n} f_{λ}^{2} = n!,

and

∑_{λ⊢n} f_{λ} = coeff of \frac{x^{n}}{n!} in e^{x+x^{2}/2}.

∗ With support from NSERC and FCAR.

We are interested in the enumeration of tableaux of height bounded by some integer k; that is to say we wish to compute the numbers

τ_{k}(n) = ∑_{h(λ)≤k} f_{λ},  (1)

as well as

T_{k}(n) = ∑_{h(λ)≤k} f_{λ}^{2}.  (2)

For example, the first few sequences τ_{k}(n) for n ≥ 1 are

τ_{2}(n) → 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462, 924, 1716, 3432, 6435, 12870, 24310, 48620, 92378, …
τ_{3}(n) → 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, …
τ_{4}(n) → 1, 2, 4, 10, 25, 70, 196, 588, 1764, 5544, 17424, 56628, 184041, 613470, 2044900, …
τ_{5}(n) → 1, 2, 4, 10, 26, 75, 225, 715, 2347, 7990, 27908, 99991, 365587, 1362310, 5159208, …
τ_{6}(n) → 1, 2, 4, 10, 26, 76, 231, 756, 2556, 9096, 33231, 126060, 488488, 1948232, 7907185, …

(These are sequences http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001405A001405, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001006A001006, http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=005817A005817, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=049401A049401, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=007579A007579 in [5].) For T_{k}(n), we have

T_{2}(n) → 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, …
T_{3}(n) → 1, 2, 6, 23, 103, 513, 2761, 15767, 94359, 586590, 3763290, 24792705, 167078577, …
T_{4}(n) → 1, 2, 6, 24, 119, 694, 4582, 33324, 261808, 2190688, 19318688, 178108704, 1705985883, …
T_{5}(n) → 1, 2, 6, 24, 120, 719, 5003, 39429, 344837, 3291590, 33835114, 370531683, 4285711539, …
T_{6}(n) → 1, 2, 6, 24, 120, 720, 5039, 40270, 361302, 3587916, 38957991, 457647966, 5763075506, …

(Sequences http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000108A000108, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A005802A005802, http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=052397A052397, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=052398A052398, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=052399A052399 in [5].)

In [2] Gessel deduces the following formulae from a result of Gordon:

y_{k}(x) := ∑_{n=0}^{∞} \frac{τ_{k}(n)x^{n}}{n!} = \{ \begin{matrix}det [J_{i−j}(x) − J_{i+j−1}(x)]_{1≤i,j≤k/2} & if k is even, \\ e^{x} det [J_{i−j}(x) − J_{i+j}(x)]_{1≤i,j≤(k−1)/2} & if k is odd, \end{matrix}

and

Y_{k}(x) := ∑_{n=0}^{∞} \frac{T_{k}(n)x^{n}}{(n!)^{2}} = det [I_{i−j}(x)]_{1≤i,j≤k} ,

where

J_{k}(x) = ∑_{n=0}^{∞} \frac{x^{2 n+k}}{n! (n + k)!}

and

I_{k}(x) = ∑_{n=0}^{∞} \frac{x^{n+k/2}}{n! (n + k)!}

If k is positive integer, we set J_{−k} := J_{k} and I_{−k} := I_{k}. The resulting expressions rapidly become unwieldy. For example,

y_{2}(x) = J_{0}(x) + J_{1}(x)

y_{3}(x) = e^{x} (J_{0}(x) − J_{2}(x))

y_{4}(x) = J_{0}(x)^{2} + J_{0}(x) J_{1}(x) + J_{0}(x) J_{3}(x) − J_{1}(x)^{2} − 2 J_{1}(x) J_{2}(x) + J_{1}(x) J_{3}(x) − J_{2}(x)^{2}

y_{5}(x) = e^{x} (J_{0}(x)^{2} − J_{0}(x) J_{2}(x) − J_{0}(x) J_{4}(x) − J_{1}(x)^{2} + 2 J_{1}(x) J_{3}(x) + J_{2}(x) J_{4}(x) − J_{3}(x)^{2})

y_{6}(x) = J_{0}(x)^{3} + J_{0}(x)^{2} J_{1}(x) + J_{0}(x)^{2} J_{3}(x) + J_{0}(x)^{2} J_{5}(x) − 2 J_{0}(x) J_{1}(x)^{2} − 2 J_{0}(x) J_{1}(x) J_{2}(x)
+ J_{0}(x) J_{1}(x) J_{3}(x) − 2 J_{0}(x) J_{1}(x) J_{4}(x) + J_{0}(x) J_{1}(x) J_{5}(x) − 2 J_{0}(x) J_{2}(x)^{2}
− 2 J_{0}(x) J_{2}(x) J_{3}(x) − J_{0}(x) J_{3}(x)^{2} + J_{0}(x) J_{3}(x) J_{5}(x) − J_{0}(x) J_{4}(x)^{2} − J_{1}(x)^{3}
+ 2 J_{1}(x)^{2} J_{2}(x) + 2 J_{1}(x)^{2} J_{3}(x) − 2 J_{1}(x)^{2} J_{4}(x) − J_{1}(x)^{2} J_{5}(x) + 2 J_{1}(x) J_{2}(x)^{2}
+ 2 J_{1}(x) J_{2}(x) J_{3}(x) + 2 J_{1}(x) J_{2}(x) J_{4}(x) − 2 J_{1}(x) J_{2}(x) J_{5}(x) + 2 J_{1}(x) J_{3}(x) J_{4}(x)
+ J_{1}(x) J_{3}(x) J_{5}(x) − J_{1}(x) J_{4}(x)^{2} − J_{2}(x)^{2} J_{3}(x) + 2 J_{2}(x)^{2} J_{4}(x) − J_{2}(x)^{2} J_{5}(x)
− 2 J_{2}(x) J_{3}(x)^{2} + 2 J_{2}(x) J_{3}(x) J_{4}(x) − J_{3}(x)^{3}

We can simplify these using properties of Bessel functions. Recalling the easily deduced relations

J_{k}(x) = J_{k−2}(x) − \frac{1}{x}(n − k − 1) J_{k−1}(x), k ≥ 2,

we get, after some computation, the much simpler expressions

y_{3}(x) = x^{−1}e^{x} J_{1}(x)

y_{4}(x) = x^{−2} (−2 x J_{0}(x)^{2} + 2 J_{0}(x) J_{1}(x) + (2 x + 1) J_{1}(x)^{2})

y_{5}(x) = x^{−4}e^{x} (−4 x^{2} J_{0}(x)^{2} + 2 x J_{0}(x) J_{1}(x) + 2 (2 x^{2} + 1) J_{1}(x)^{2})

\begin{matrix}y_{6}(x) = & x^{−6} (−4 x^{2} (4 x − 3) J_{0}(x)^{3} − 4 x (4 x^{2} − 3 x + 6) J_{0}(x)^{2} J_{1}(x) \\ + 4 (4 x^{3} − x^{2} + 3) J_{0}(x) J_{1}(x)^{2} + 4 (4 x^{3} − x^{2} + 5 x + 1) J_{1}(x)^{3}) \end{matrix}

Similarly

Y_{2}(x) = I_{0}(x)^{2} − I_{1}(x)^{2}

Y_{3}(x) = x^{−1} (2 \sqrt{x} I_{0}(x)^{2} I_{1}(x) − I_{0}(x) I_{1}(x)^{2} − 2 \sqrt{x} I_{1}(x)^{3})

\begin{matrix}Y_{4}(x) = & x^{−3}( − 4 x^{2} I_{0}(x)^{4} + 8 x \sqrt{x} I_{0}(x)^{3} I_{1}(x) + 4 x (2 x − 1) I_{0}(x)^{2} I_{1}(x)^{2} \\ − 8 x \sqrt{x} I_{0}(x) I_{1}(x)^{3} − x (4 x − 1) I_{1}(x)^{4}) \end{matrix}

A theoretical argument (see [2]) shows that the generating functions y_{k}(x) and Y_{k}(x) are D-finite. That is to say, they satisfy linear differential equations with polynomial coefficients. In fact, it is well known and classical that one can translate such linear differential equations into recurrences with polynomial coefficients. More precisely, a P -recurrence for a sequence a_{n} is one of the form

p_{0}(n) a_{n} + p_{1}(x) a_{n−1} + … + p_{k}(n) a_{n−k} = q(n),

where all p_{i}(n), 1 ≤ i ≤ k, and q(n) are polynomials in n. We say that a sequence is P -recursive if it satisfies a P -recurrence. The class of P -recursive sequences is closed under point-wise products. Since 1/n! is easily seen to be P -recursive, it follows that, if a_{n} is P -recursive, then so are a_{n}/n! and a_{n}/n!^{2}. The algorithmic translation from D-finite to P -recursive (and back) has been implemented in the package GFUN in Maple (see [4]), which also contains many other nice tools for handling recurrences and generating functions.

Computer experiments made by Krob, Favreau and the first author led to conjectures (see [1]) for an explicit form for P -recurrences for τ_{h}(n) and T_{h}(n). These conjectures can be easily (and automatically) reformulated as linear differential equations for y_{k}(x) and Y_{k}(x). We first observe that it is not hard to show the existence of a linear differential equation of order bounded by

ℓ(k) := ⌊\frac{k}{2} ⌋ + 1

with polynomial coefficients, admitting y_{k}(x) as a solution. In fact, this follows readily from the following proposition.

Proposition 1. Let V_{k} denote the vector space over the field C(x) of rational functions in x spanned by y_{k}(x) and all its derivatives. Then

dim V_{k} ≤ ℓ(k).

Proof. Setting n := ℓ(k) − 1, it is clear from our previous discussion that y_{k} lies in the span W_{k} of the set of ℓ(k) elements given by

{J_{0}(x)^{m} J_{1}(x)^{n−m} | 0 ≤ m ≤ n }

if k is even, and by

{e^{x} J_{0}(x)^{m} J_{1}(x)^{n−m} | 0 ≤ m ≤ n }

if k is odd. W_{k} is clearly closed under differentiation, since we easily see that

\begin{aligned}\frac{d}{dx}J_{0}(x) & = 2 J_{1}(x), \\ \frac{d}{dx}J_{1}(x) & = 2 J_{0}(x) − \frac{1}{x} J_{1}(x), \end{aligned}  (1)

from which we deduce that

\frac{d}{dx}J_{0}(x)^{a} J_{1}(x)^{b} = \frac{2 a}{x} J_{0}(x)^{a−1} J_{1}(x)^{b+1} x + 2 b J_{0}(x)^{a+1} J_{1}(x)^{b−1} − \frac{b}{x} J_{0}(x)^{a} J_{1}(x)^{b},  (2)

as well as a similar expression for the derivative of e^{x} J_{0}(x)^{a} J_{1}(x)^{b}. Thus V_{k} is contained in W_{k}, and hence its dimension is bounded by ℓ(k).

Setting for the moment n := ℓ(k) and y := y_{k}(x), it clearly follows from the above proposition that

y, y′, y′′, …, y^{(n)}

are linearly dependent, hence y_{k}(x) satisfies a homogeneous linear differential equation of order (at most) ℓ(k) with polynomial coefficients (in x). However, it appears that a stronger result holds.

Conjecture (Bergeron–Favreau–Krob, [1]). For each k, there are polynomials p_{m}(x) of degree at most ℓ − 1 such that y_{k}(x) is a solution of

∑_{m=0}^{ℓ} p_{m}(x)y^{(m)} = 0,

where ℓ = ℓ(k). Moreover, for m ≥ 1, p_{m}(x) = q_{m}(x) x^{m−1}, and p_{ℓ}(x) = x^{ℓ−1}.

The first few cases for y_{k}(x) are^{∗}

∗ Here → means “is a solution of”. y_{2}(x) → x y′′ + 2 y′ − 2 (2 x + 1) y = 0 y_{3}(x) → x y′′ − (2 x − 3) y′ − 3 (x + 1) y = 0 y_{4}(x) → x^{2} y′′′ + 10 x y′′ − 4 (4 x^{2} + 2 x − 5) y′ − 4 (8 x + 5) y = 0 y_{5}(x) → x^{2} y′′′ − (3 x − 13) x y′′ − (13 x^{2} + 26 x − 35) y′ + 5 (3 x^{2} − 7 x − 7) y = 0 Equating coefficients of x^{n}/n! on both hand sides of these differential equations, one finds that they are equivalent to the recurrences (n + 1) τ_{2}(n) − 2 τ_{2}(n − 1) − 4 (n − 1) τ_{2}(n − 2) = 0 (n + 2) τ_{3}(n) − (2 n + 1) τ_{3}(n − 1) − 3 (n − 1) τ_{3}(n − 2) = 0 (n + 3) (n + 4) τ_{4}(n)−16 (n − 1) τ_{4}(n − 2) n − (8 n + 12) τ_{4}(n − 1) = 0 (n+4) (n+6) τ_{5}(n)−(3 n^{2} + 17 n + 15) τ_{5}(n−1)−(n−1)(13 n+9) τ_{5}(n−2)+15 (n−1) (n−2) τ_{5}(n−3) = 0

Up to now, only these recurrences (that is, for k ≤ 5), had been implicitly known (see [3]). However, using the simplified expressions for y_{k}(x) given here, and a reformulation in term of linear differential equations (with the help of GFUN [4]) we have been able to check (in the form of a computer algebra proof) that the conjecture above is true for k ≤ 11, from which it follows that the corresponding recurrences hold. This computer verification simply uses the derivation rules (1) for J_{0}(x) and J_{1}(x) to simplify the expressions obtained by substitution of Gessel’s formulae in the following differential equations.

\begin{aligned}y_{6}(x) → & x^{3} y^{(4)} + 28 x^{2} y′′′ − 10 (4 x^{2} + 2 x − 23) x y′′ \\ & − 4 (108 x^{2} + 61 x − 135) y′ + 36 (2 x + 5) (2 x^{2} − 3 x − 3) y = 0 \end{aligned}

\begin{aligned}y_{7}(x) → & x^{3} y^{(4)} − 2 (2 x − 17) x^{2} y′′′ − (34 x^{2} + 102 x − 343) x y′′ \\ & + (76 x^{3} − 450 x^{2} − 686 x + 1001) y′ + 7 (15 x^{3} + 74 x^{2} − 143 x − 143) y = 0 \end{aligned}

\begin{aligned}y_{8}(x) → & x^{4} y^{(5)} + 60 x^{3} y^{(4)} − 2 (40 x^{2} + 20 x − 619) x^{2} y′′′ − 4 (608 x^{2} + 331 x − 2567) x y′′ \\ & + 8 (128 x^{4} + 128 x^{3} − 2480 x^{2} − 1527 x + 3536) y′ + 128 (64 x^{3} + 72 x^{2} − 286 x − 221) y = 0 \end{aligned}

y_{9}(x) → x^{4} y^{(5)} − 5 (x − 14) x^{3} y^{(4)} − (70 x^{2} + 280 x − 1693) x^{2} y′′′
+ (230 x^{3} − 2492 x^{2} − 5079 x + 16535) x y′′
+ (789 x^{4} + 5544 x^{3} − 24073 x^{2} − 33070 x + 53865) y′
− 27 (35 x^{4} − 274 x^{3} − 1017 x^{2} + 1995 x + 1995) y = 0
y_{10}(x) → x^{5} y^{(6)} + 110 x^{4} y^{(5)} − 2 (70 x^{2} + 35 x − 2269) x^{3} y^{(4)}
− 4 (2268 x^{2} + 1211 x − 21752) x^{2} y′′′
+ 4 (1036 x^{4} + 1036 x^{3} − 48033 x^{2} − 27900 x + 191477) x y′′
+ 8 (14300 x^{4} + 15542 x^{3} − 185404 x^{2} − 121352 x + 303875) y′
− 200 (72 x^{5} + 108 x^{4} − 3262 x^{3} − 3987 x^{2} + 14960 x + 12155) y = 0

y_{11}(x) → x^{5} y^{(6)} − (6 x − 125) x^{4} y^{(5)} − (125 x^{2} + 625 x − 5873) x^{3} y^{(4)}
+ 2 (270 x^{3} − 4611 x^{2} − 11746 x + 64252) x^{2} y′′′
+ (3319 x^{4} + 30166 x^{3} − 223422 x^{2} − 385512 x + 1293125) x y′′
− (7734 x^{5} − 104329 x^{4} − 493828 x^{3} + 1987124 x^{2} + 2586250 x − 4697275) y′
− 11 (945 x^{5} + 11343 x^{4} − 62023 x^{3} − 204012 x^{2} + 427025 x + 427025) y = 0

However, these verifications rapidly become (computer) time consuming. For example, with k = 11, we have to substitute in this last differential equation the following expression

y_{11}(x) = \frac{138240 e^{x}}{x^{25}} ( − 14 (32 x^{6} + 177 x^{4} + 198 x^{2} − 72) x^{5} J_{0}(x)^{5}
+ 8 (16 x^{8} + 256 x^{6} + 825 x^{4} + 585 x^{2} − 495) x^{4} J_{1}(x) J_{0}(x)^{4}
+ 4 (192 x^{8} + 833 x^{6} + 495 x^{4} + 135 x^{2} + 1440) x^{3} J_{1}(x)^{2} J_{0}(x)^{3}
− (256 x^{10} + 3648 x^{8} + 10799 x^{6} + 9690 x^{4} + 1980 x^{2} + 3600) x^{2} J_{1}(x)^{3} J_{0}(x)^{2}
− 5 (64 x^{10} + 190 x^{8} − 77 x^{6} + 114 x^{4} + 504 x^{2} − 144) x J_{1}(x)^{4} J_{0}(x)
+ (128 x^{12} + 1632 x^{10} + 4557 x^{8} + 5482 x^{6} + 4158 x^{4} + 2052 x^{2} + 72) J_{1}(x)^{5})

and simplify. Clearly we could go on to larger cases, but the point seems to be made that the conjectures are reasonable.

Similar considerations for the enumeration of pairs of tableaux, with the following differential equations, settle the corresponding conjectures for the cases k ≤ 7:

Y_{2}(x) → x^{2} y′′′ + 4 x y′′ − 2 (2 x − 1) y′ − 2 y = 0

Y_{3}(x) → x^{3} y^{(4)} + 10 x^{2} y′′′ − (10 x − 23) x y′′ − (32 x − 9) y′ + 9 (x − 1) y = 0

\begin{aligned}Y_{4}(x) → & x^{4} y^{(5)} + 20 x^{3} y^{(4)} − 2 (10 x − 59) x^{2} y′′′ − 2 (91 x − 110) x y′′ \\ & + 4 (16 x^{2} − 87 x + 20) y′ + 16 (8 x − 5) y = 0 \end{aligned}

\begin{aligned}Y_{5}(x) → & x^{5} y^{(6)} + 35 x^{4} y^{(5)} − 7 (5 x − 59) x^{3} y^{(4)} − 2 (336 x − 979) x^{2} y′′′ + (259 x^{2} − 3650 x + 3383) x y′′ \\ & + (1917 x^{2} − 5708 x + 1225) y′ − 25 (9 x^{2} − 93 x + 49) y = 0 \end{aligned}

Y_{6}(x) → x^{6} y^{(7)} + 56 x^{5} y^{(6)} − 28 (2 x − 41) x^{4} y^{(5)} − 4 (483 x − 2684) x^{3} y^{(4)}
+ 4 (196 x^{2} − 5480 x + 11543) x^{2} y′′′ + 8 (1686 x^{2} − 11941 x + 9830) x y′′
− 4 (576 x^{3} − 14931 x^{2} + 34438 x − 7290) y′ − 72 (144 x^{2} − 821 x + 405) y = 0
Y_{7}(x) → x^{7} y^{(8)} + 84 x^{6} y^{(7)} − 42 (2 x − 65) x^{5} y^{(6)} − 2 (2352 x − 21881) x^{4} y^{(5)}
+ 3 (658 x^{2} − 31606 x + 121455) x^{3} y^{(4)} + 2 (31986 x^{2} − 424260 x + 754183) x^{2} y′′′
− (12916 x^{3} − 648834 x^{2} + 3329230 x − 2610671) x y′′
− (175704 x^{3} − 2292734 x^{2} + 4684008 x − 1002001) y′
+ 49 (225 x^{3} − 9630 x^{2} + 42313 x − 20449) y = 0

2. Acknowledgments

The Maple package gfun (available as a shared library) was used extensively in the elaboration of the conjectures and results in this note.

3. References

[1] F. Bergeron, L. Favreau and D. Krob, Conjectures on the Enumeration of Tableaux of Bounded Height, Discrete Math., 139, (1995), 463–468.

[2] I. Gessel, Symmetric Functions and P-Recursiveness, Jour. of Comb. Th., Series A, 53, 1990, 257–285.

[3] D. Gouyou Beauchamps, Codages par des mots et des chemins: problèmes combinatoires et algorithmiques, Ph. D. thesis, University of Bordeaux I, 1985.

[4] B. Salvy and P. Zimmermann, GFUN: A maple Package for the Manipulation of Generating Functions in one Variable, ACM Trans. in Math. Software, 20, 1994, pages 163–177.

[5] N.J.A Sloane, The On-Line Encyclopedia of Integer Sequences, published electornically at http://www.research.att.com/ njas/sequences/http://www.research.att.com/∼njas/sequences/. See also N.J.A Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995.

(Concerned with sequence http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000108A000108, http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001006A001006, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001405A001405, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=005802A005802, http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=005817A005817, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=007579A007579, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=049401A049401, http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=052397A052397, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=052398A052398, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=052399A052399.)

Received Nov. 10, 1999; published in Journal of Integer Sequences March 15, 2000.

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