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On the difference of successive Gaussian polynomials
Journal
of Statistical
Planning
and Inference
19
34 (1993) 19-22
North-Holland
On the difference of successive Gaussian polynomials George E. Andrews* Department
of Mathematics.Pennsylvania
Received 21 August
State University, University Park, PA 16802, USA
1990; revised manuscript
Abstract:
In this paper
Gaussian
polynomials,
we provide
The Gaussian
a partition-theoretic
1991
interpretation
of the difference
of two successive
namely q’([“;‘] - r?fi).
AMS Subject ClassiJication: Key words; Gaussian
received 26 March
1 lP68.
polynomial;
partitions;
polynomials M
[L] are defined 1
n’
j=l
successive ranks; Frobenius
symbols.
by
_qNfl-j)
if O
(1-qj)
(1) if M
0
or M>N.
Our object in this paper is to relate differences of Gaussian polynomials to partitions through the use of Frobenius symbols (Andrews, 1984, Section 2). For any partition rc, its Frobenius symbol is constructed as follows: In the Ferrers graph of rr, delete the main diagonal (of say r nodes); then create a two-line array of integers wherein the upper row consists of the cardinalities of the r rows in the Ferrers graph to the right of the main diagonal and the lower row consists of the cardinalities of the r columns in the Ferrers graph below the main diagonal. For example, if rr is the partition 5 + 4 + 4 + 2, then the Ferrers graph of rr is as depicted in Figure 1 and the Frobenius symbol is (i i A). In this way we see that there is a bijection between partitions of integers and equi-length two-rowed arrays of nonnegative integers with strict decrease on each row; the latter are the Frobenius symbols. We shall prove the following theorem.
Correspondence to: G.E. Andrew, Park, PA 16802, USA. Partially
supported
0378-3758/93/$06.00
by National
0
Department
of Mathematics,
Science Foundation
1993-Elsevier
Science
Grant
Publishers
Pennsylvania
State University,
DMS 8702695-03.
B.V. All rights
reserved
University
G.E. Andrews
20
/ Gaussian polynomials
Fig. 1.
For 0
Theorem.
is the generating function for partitions n whose Frobenius symbols
satisfya,
b,
andfor
By setting j = N in our Theorem, Catalan numbers: Corollary
l
ai-bi
interpretation
of the q-
1. The q-Catalan number
(l-q) (I-qN+l)
2N c N 1
is the generating function for all partitions whose Frobenius symbols have all entries
if we take N=4
in Corollary
1, the resulting
polynomial
is
1 +q2+q3+2q4+q5+2q6+q7+2q8+q9+q10+q12 and the relevant Table 1. By letting N+
partitions
together
COin Corollary
with their Frobenius
symbols
1, we see that the q-Catalan
are as given in
number
converges
to
fgj=2
Consequently
l-qi’ we obtain:
Corollary 2. The number of partitions of n without ones equals the number of partitions of n with Frobenius symbols having strictly increasing columns.
G.E. Andrews / Gaussian polynomials
Table
21
1 Frobenius
Partition
”
n
Partition
Frobenius symbol
symbol 2
1+1
(?I
7
2+2+2+1
(::I
3
1+1+1
&
8
2+2+2+2
c:;,
4
2+1+1
c:,
3+2+2+1
(:Y)
1+1+1+1
(PI
9
3+2+2+2
(:;I
5
2+1+1+1
c:,
10
3+3+2+2
c:,
6
2+2+2
(::I
12
3+3+3+3
c::?)
3+1+1+1
(:I
For example, 9,
when n = 9 the eight partitions
of the first type are
7+2,
4+3+2,
6+3,
while the eight partitions
5+4,
5+2+2,
of the second
4+1+1+1+1+1,
3+2+2+2
type are
3+2+2+2,
3+2+2+1+1,
3+1+1+1+1+1+1,2+2+2+2+1, 2+1+1+1+1+1+1+1,
3+3+3,
2+2+2+1+1+1, 1+1+1+1+1+1+1+1+1.
Before we proceed we should note that others have found enumerative interpretations of the differences of Gaussian polynomials. Dennis Stanton informed me that Susanna Fishel has obtained a related but somewhat different interpretation of this difference based upon identities of Macdonald (1979), page 130. Also Butler (1987) has found a much broader interpretation in the theory of finite Abelian groups. The reason the following proof is so short is that the central enumerative observation is proved in an old paper on successive ranks (Andrews, 1972). Proof of Theorem.
We first note that
(by Andrews,
Now [“+;-‘I
1976, p. 35, (3.3.4)
is the generating
function
and (3.3.3))
for all partitions
with at most j parts and
G. E. Andrews / Gaussian polynomials
22
each part
References Andrews,
G.E.
Andrews,
G.E. (1976). In: G.-C.
(1972). Sieves in the theory
of partitions.
and Its Applications. Vol. 2 Addison-Wesley, London
J. Math. 94, 1214-1230.
Reading,
MA. (Reissued:
Cambridge
University
Press,
and New York).
Andrews,
G.E. (1984). Generalized
Frobenius
Bressoud,
D.M.
of the partitions
Butler,
Amer.
Rota, Ed., The Theory of Partitions, the Encyclopedia of Mathematics
L.M.
(1980). Extension
(1987). A unimodality
Mem. Amer. Math. Sot. 49 (301), iv+44
partitions.
pp.
sieve. J. Number Theory 12, 87-100.
result in the enumeration
of subgroups
of a finite Abelian
group.
Proc. Amer. Math. Sot. 101, 771-775. Furlinger, Guttmann, Phys.
J. and J. Hofbauer A.J.
(1985). q-Catalan
and M. Hirschhorn
numbers.
(1984). Comment
J. Combin. Theory Ser. A 40, 248-264.
on the number
of spiral self-avoiding
walks.
J.
A 17, 3613-3614.
Macdonald, I.G. (1979). Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford. MacMahon, P.A. (1976). Coliected Papers. Vol. 1 (G.E. Andrews, Ed.). MIT Press, Cambridge, MA. Roselle, D.P. (1974). A combinatorial problem involving q-Catalan numbers. Notices Amer. Math. Sot. 21 (A), 609.
of Statistical
Planning
and Inference
19
34 (1993) 19-22
North-Holland
On the difference of successive Gaussian polynomials George E. Andrews* Department
of Mathematics.Pennsylvania
Received 21 August
State University, University Park, PA 16802, USA
1990; revised manuscript
Abstract:
In this paper
Gaussian
polynomials,
we provide
The Gaussian
a partition-theoretic
1991
interpretation
of the difference
of two successive
namely q’([“;‘] - r?fi).
AMS Subject ClassiJication: Key words; Gaussian
received 26 March
1 lP68.
polynomial;
partitions;
polynomials M
[L] are defined 1
n’
j=l
successive ranks; Frobenius
symbols.
by
_qNfl-j)
if O
(1-qj)
(1) if M
0
or M>N.
Our object in this paper is to relate differences of Gaussian polynomials to partitions through the use of Frobenius symbols (Andrews, 1984, Section 2). For any partition rc, its Frobenius symbol is constructed as follows: In the Ferrers graph of rr, delete the main diagonal (of say r nodes); then create a two-line array of integers wherein the upper row consists of the cardinalities of the r rows in the Ferrers graph to the right of the main diagonal and the lower row consists of the cardinalities of the r columns in the Ferrers graph below the main diagonal. For example, if rr is the partition 5 + 4 + 4 + 2, then the Ferrers graph of rr is as depicted in Figure 1 and the Frobenius symbol is (i i A). In this way we see that there is a bijection between partitions of integers and equi-length two-rowed arrays of nonnegative integers with strict decrease on each row; the latter are the Frobenius symbols. We shall prove the following theorem.
Correspondence to: G.E. Andrew, Park, PA 16802, USA. Partially
supported
0378-3758/93/$06.00
by National
0
Department
of Mathematics,
Science Foundation
1993-Elsevier
Science
Grant
Publishers
Pennsylvania
State University,
DMS 8702695-03.
B.V. All rights
reserved
University
G.E. Andrews
20
/ Gaussian polynomials
Fig. 1.
For 0
Theorem.
is the generating function for partitions n whose Frobenius symbols
satisfya,
b,
andfor
By setting j = N in our Theorem, Catalan numbers: Corollary
l
ai-bi
interpretation
of the q-
1. The q-Catalan number
(l-q) (I-qN+l)
2N c N 1
is the generating function for all partitions whose Frobenius symbols have all entries
if we take N=4
in Corollary
1, the resulting
polynomial
is
1 +q2+q3+2q4+q5+2q6+q7+2q8+q9+q10+q12 and the relevant Table 1. By letting N+
partitions
together
COin Corollary
with their Frobenius
symbols
1, we see that the q-Catalan
are as given in
number
converges
to
fgj=2
Consequently
l-qi’ we obtain:
Corollary 2. The number of partitions of n without ones equals the number of partitions of n with Frobenius symbols having strictly increasing columns.
G.E. Andrews / Gaussian polynomials
Table
21
1 Frobenius
Partition
”
n
Partition
Frobenius symbol
symbol 2
1+1
(?I
7
2+2+2+1
(::I
3
1+1+1
&
8
2+2+2+2
c:;,
4
2+1+1
c:,
3+2+2+1
(:Y)
1+1+1+1
(PI
9
3+2+2+2
(:;I
5
2+1+1+1
c:,
10
3+3+2+2
c:,
6
2+2+2
(::I
12
3+3+3+3
c::?)
3+1+1+1
(:I
For example, 9,
when n = 9 the eight partitions
of the first type are
7+2,
4+3+2,
6+3,
while the eight partitions
5+4,
5+2+2,
of the second
4+1+1+1+1+1,
3+2+2+2
type are
3+2+2+2,
3+2+2+1+1,
3+1+1+1+1+1+1,2+2+2+2+1, 2+1+1+1+1+1+1+1,
3+3+3,
2+2+2+1+1+1, 1+1+1+1+1+1+1+1+1.
Before we proceed we should note that others have found enumerative interpretations of the differences of Gaussian polynomials. Dennis Stanton informed me that Susanna Fishel has obtained a related but somewhat different interpretation of this difference based upon identities of Macdonald (1979), page 130. Also Butler (1987) has found a much broader interpretation in the theory of finite Abelian groups. The reason the following proof is so short is that the central enumerative observation is proved in an old paper on successive ranks (Andrews, 1972). Proof of Theorem.
We first note that
(by Andrews,
Now [“+;-‘I
1976, p. 35, (3.3.4)
is the generating
function
and (3.3.3))
for all partitions
with at most j parts and
G. E. Andrews / Gaussian polynomials
22
each part
References Andrews,
G.E.
Andrews,
G.E. (1976). In: G.-C.
(1972). Sieves in the theory
of partitions.
and Its Applications. Vol. 2 Addison-Wesley, London
J. Math. 94, 1214-1230.
Reading,
MA. (Reissued:
Cambridge
University
Press,
and New York).
Andrews,
G.E. (1984). Generalized
Frobenius
Bressoud,
D.M.
of the partitions
Butler,
Amer.
Rota, Ed., The Theory of Partitions, the Encyclopedia of Mathematics
L.M.
(1980). Extension
(1987). A unimodality
Mem. Amer. Math. Sot. 49 (301), iv+44
partitions.
pp.
sieve. J. Number Theory 12, 87-100.
result in the enumeration
of subgroups
of a finite Abelian
group.
Proc. Amer. Math. Sot. 101, 771-775. Furlinger, Guttmann, Phys.
J. and J. Hofbauer A.J.
(1985). q-Catalan
and M. Hirschhorn
numbers.
(1984). Comment
J. Combin. Theory Ser. A 40, 248-264.
on the number
of spiral self-avoiding
walks.
J.
A 17, 3613-3614.
Macdonald, I.G. (1979). Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford. MacMahon, P.A. (1976). Coliected Papers. Vol. 1 (G.E. Andrews, Ed.). MIT Press, Cambridge, MA. Roselle, D.P. (1974). A combinatorial problem involving q-Catalan numbers. Notices Amer. Math. Sot. 21 (A), 609.