Volume 16, Number 1, Pages 117–127 ISSN 1715-0868

GEOMETRIC POLYNOMIALS AND INTEGER PARTITIONS

MIRCEA MERCA

Abstract. In this paper, we show that the geometric polynomials can be expressed as sums over integer partitions in two different ways. New formulas involving geometric numbers, Bernoulli numbers, and Genocchi numbers are derived in this context.

1. Introduction The geometric polynomials (also known as Fubini polynomials) are de- fined as follows (see [22, 24, 26]): n X n (1.1) ω (x) = k!xk, n k k=0 n where k are the Stirling numbers of the second kind. Recall that the Stirling numbers of the second kind count the number of ways to partition a set of n objects into k nonempty subsets. In terms of partitions of an n n-set, k k! is the number of distinct ordered partitions with k subsets. 3 For example, 2 = 3 because the set {1, 2, 3} can be partitioned into two subsets in three ways: {1, 2}, {3} , {1, 3}, {2} and {1}, {2, 3} . The ordered partitions {1, 2}, {3} , {3}, {1, 2} , {1, 3}, {2} , {2}, {1, 3} ,   3 {1}, {2, 3} , and {2, 3}, {1} are counted by 2 2! = 6. The first few geometric polynomials are:

ω0(x) = 1,

ω1(x) = x, 2 ω2(x) = x + 2x , 2 3 ω3(x) = x + 6x + 6x , 2 3 4 ω4(x) = x + 14x + 36x + 24x .

Received by the editors September 29, 2020, and in revised form November 24, 2020. 2010 Mathematics Subject Classification. 11B73, 11B65, 11P81, 05A17, 05A19. Key words and phrases. geometric polynomials, geometric numbers, Bernoulli numbers, Genocchi numbers, partitions.

This work is licensed under a Creative Commons “Attribution- NoDerivatives 4.0 International” license.

117 118 MIRCEA MERCA

The geometric polynomials first appeared in Euler’s book [13, Part 2, § d
m 172]. They describe the action of the derivative operator x dx , m = 0, 1, 2,... on the function 1/(1 − x), m ∞  d  1 X 1  1  x = kmxk = F , |x| < 1. dx 1 − x 1 − x m 1 − x k=0 The use of geometric polynomials by Euler was pointed out by K. Boyadzhiev in [6] and H. Gould mentions these polynomials in [14]. Gould refers to the book of I. J. Schwatt [24]. In the last two decades, the geometric polynomials have received consid- erable attention as an effective tool in different topics in analysis, combina- torics and number theory [4, 5, 7, 8, 10, 12, 11, 16, 17, 19, 20, 22]. In [10] the authors considered Euler–Seidel matrices method and obtained some fundamental properties of the geometric polynomials as the following linear recurrence relations: n X n + 1 (1.2) ω (x) = x ω (x), n+1 k k k=0 n x X n + 1 ω (x) = ω (x) + ω (x)
, n+1 1 + x k k k+1 k=0 d ω (x) = x ω (x) + xω (x)
. n+1 dx n n In this paper, using the exponential generating functions of the geometric polynomials [4], i.e., ∞ X zn 1 (1.3) ω (x) = , n n! 1 − x(ez − 1) n=0 we shall establish formulas for ωn(x) or ωn−1(x) as sums over all the un- restricted integer partitions of n. Recall [1] that a partition of a positive integer n is a weakly decreasing sequence of positive integers

λ1 > λ2 > ··· > λk > 0 whose sum is n,

λ1 + λ2 + ··· + λk = n. The positive integers in the sequence are called parts. To make formulas more concise, we pad the sequence of parts with zeros to obtain n nonneg- ative parts. Theorem 1.1. For n > 0, λ1
λ2
··· λn
n! X λ2 λ3 0 λ1 ωn(x) = x . 1λ1 2λ2 ··· nλn λ1+λ2+···+λn=n λ1>λ2>...>λn>0 GEOMETRIC POLYNOMIALS 119

For example, the case n = 4 of this theorem reads as follows 4
4! 3
1
4! 2
2
4! ω (x) = 0 x4 + 1 0 x3 + 2 0 x2 4 14 1321 1222 2
1
1
4! 1
1
1
1
4! + 1 1 0 x2 + 1 1 1 0 x 122131 11213141 = 24x4 + 36x3 + 6x2 + 8x2 + x = 24x4 + 36x3 + 14x2 + x, because the partitions in question are: (1.4) 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1. Theorem 1.2. For n > 1,

λ1
λ2
λn
··· n! λ1 1 X λ2 λ3 0 x ωn−1(x) = · λ λ λ . x + 1 1 1 2 2 ··· n n λ1 λ1+λ2+···+λn=n λ1>λ2>...>λn>0 For example, using (1.4) and Theorem 1.2, we can write:

1 4
4! x4 3
1
4! x3 2
2
4! x2 ω (x) = 0 + 1 0 + 2 0 3 x + 1 14 4 1321 3 1222 2 ! 2
1
1
4! x2 1
1
1
1
4! + 1 1 0 + 1 1 1 0 x 122131 2 11213141 6x4 + 12x3 + 7x2 + x = x + 1 (x + 1)(6x3 + 6x2 + x) = x + 1 = 6x3 + 6x2 + x. Related to Theorems 1.1 and 1.2, we remark that the coefficient λ1
λ2
··· λn
n! λ2 λ3 0 1λ1 2λ2 ··· nλn is the number of preferential arrangements associated with an integer parti- tion of n and can be seen in the On-Line Encyclopedia of Integer Sequence [25, A049019]. For x = 1 in (1.1), we get nth geometric number (ordered Bell number or Fubini number): n X n ω = ω (1) = k!. n n k k=0 These numbers were called Fubini numbers by Louis Comtet, because they count the number of different ways to rearrange the ordering of sums or integrals in Fubini’s theorem [22]. On the other hand, the nth geometric number counts the distinct ordered partitions of an n-set. These numbers 120 MIRCEA MERCA have been studied by many authors. Various characterizations of these num- bers can be found in the literature [2, 3, 9, 10, 15, 18, 21, 23, 27]. Taking into account Theorems 1.1 and 1.2, we can write the following decompositions of the geometric numbers. Corollary 1.3. For n > 0, λ1
λ2
··· λn
X λ2 λ3 0 ωn = · n!. 1λ1 2λ2 ··· nλn λ1+λ2+···+λn=n λ1>λ2>...>λn>0 Corollary 1.4. For n > 1, λ1
λ2
··· λn
X λ2 λ3 0 n! ωn−1 = λ λ λ . 1 1 2 2 ··· n n 2λ1 λ1+λ2+···+λn=n λ1>λ2>...>λn>0 Using (1.1) and Theorem 1.1, we easily deduce the following formula for the Stirling numbers of the second kind as a sum over the partitions of n in which the largest part has size k. Corollary 1.5. For n, k > 0,

k
λ2
λn
n X ··· n! = λ2 λ3 0 · . k 1k2λ2 ··· nλn k! k+λ2+···+λn=n k>λ2>...>λn>0 According to [17, Theorem 1.2] and [4, eq. (3.29)], we have the following relations for n > 0: Z 0 (1.5) ωn(x)dx = Bn −1 and  1 G (1.6) ω − = n+1 , n 2 n + 1 where Bn is the nth Bernoulli number and Gn is the nth Genocchi num- ber. Recall that these numbers are defined by the exponential generating functions ∞ X zn z B = , |z| < 2π n n! ez − 1 n=0 and ∞ X zn 2z G = , |z| < π. n n! ez + 1 n=0 Over the years, the works of Bernoulli numbers and Genocchi numbers and their combinatorial relations have received much attention. Using (1.5), (1.6), and Theorem 1.1, we derive new formulas for Bn and Gn+1 as sums over all the integer partitions of n. GEOMETRIC POLYNOMIALS 121

Corollary 1.6. For n > 0,

λ1
λ2
λn
n! ··· X λ1 λ2 λ3 0 Bn = (−1) λ λ λ λ1 + 1 1 1 2 2 ··· n n λ1+λ2+···+λn=n λ1>λ2>...>λn>0 and λ1
λ2
λn
(n + 1)! ··· X λ1 λ2 λ3 0 Gn+1 = (−1) . 2λ1 1λ1 2λ2 ··· nλn λ1+λ2+···+λn=n λ1>λ2>...>λn>0 The case n = 4 of this corollary reads as follows: 4! 1 4! 3 4! 1 4! 2 4! 1 B = · − · + · + · − · 4 5 14 4 1321 3 1222 4 122131 2 11213141 24 8 1 = − 9 + 2 + − 5 3 2 1 = − 30 and 5! 1 5! 3 5! 1 5! 2 5! 1 G = · − · + · + · − · 5 24 14 23 1321 22 1222 22 122131 21 11213141 15 45 15 5 = − + + 10 − 2 2 2 2 = 0. Another decomposition for the nth Genocchi number can be easily ob- tained if we consider Theorem 1.2 and the relation (1.6). Corollary 1.7. For n > 1,

λ1
λ2
λn
n · n! ··· X λ1 λ2 λ3 0 Gn = (−1) λ −1 λ λ λ . 2 1 · λ1 1 1 2 2 ··· n n λ1+λ2+···+λn=n λ1>λ2>...>λn>0 For example, the case n = 4 of Corollary 1.7 can be written as: 4 · 4! 1 4 · 4! 3 4 · 4! 1 G = · − · + · 4 23 · 4 14 22 · 3 1321 21 · 2 1222 4 · 4! 2 4 · 4! 1 + · − · 21 · 2 122131 20 · 1 11213141 = 3 − 12 + 6 + 8 − 4 = 1. The rest of this paper is organized as follows. In the next two sections we will prove Theorems 1.1 and 1.2. In Section 4 we will provide a de- composition of the geometric polynomial ωn+1(x) as a sum over the weak compositions of n into exactly 3 parts. 122 MIRCEA MERCA

2. Proof of Theorem 1.1 Using the generating functions for the geometric polynomials (1.3) and the exponential series ∞ X zn (2.1) ez = , |z| < 1, n! n=0 we can write ∞ X zn ω (x) n n! n=0 1 = 1 − x(ez − 1) ∞ !−1 X zn = 1 − x n! n=1 ∞ ∞ !j X X zn = 1 + x n! j=1 n=1   ∞ ∞   n X j X  X t1 + t2 + ··· + tn Y 1  n = 1 + x  t  z  t1, t2, . . . , tn i! i  j=1 n=1 t1+2t2+···+ntn=n i=1 t1+t2+···+tn=j ∞   X X t1 + t2 + ··· + tn = 1 + xt1+t2+···+tn × t1, t2, . . . , tn n=1 t1+2t2+···+ntn=n n ! Y 1 × zn. iti+t2+···+tn i=1 Thus we deduce that ω (x) t + t + ··· + t  n 1 n X t1+t2+···+tn 1 2 n Y = x t +t +···+t . n! t1, t2, . . . , tn i i 2 n t1+2t2+···+ntn=n i=1

We see that this decomposition of ωn(x)/(n!) is a sum over all the partitions of n. Let λ = (λ1, λ2, . . . , λn) be the conjugate partition of the partition

t1 + 2t2 + ··· + ntn = n.

It is clear that λi = ti + ti+1 + ··· + tn. In this way, we obtain

λ1
λ2
λn
ωn(x) X ··· = λ2 λ3 0 xλ1 . n! 1λ1 2λ2 ··· nλn λ1+λ2+···+λn=n λ1>λ2>···>λn>0 This concludes the proof. GEOMETRIC POLYNOMIALS 123

3. Proof of Theorem 1.2 Taking into account the exponential series (2.1) and the logarithmic series

∞ X (−1)n−1 ln(1 + z) = zn, |z| < 1, n n=1 we can write d ln 1 − x(ez − 1)
dz ∞ ! d X zn = ln 1 − x dz n! n=1 ∞ ∞ !j d X xj X zn = − dz j n! j=1 n=1   ∞ j ∞   n  ti d X x X  X t1 + t2 + ··· + tn Y 1  n = −   z dz j  t1, t2, . . . , tn i!  j=1 n=1 t1+t2+···+tn=j i=1 t1+2t2+···+ntn=n ∞ t1+t2+···+tn   d X X x t1 + t2 + ··· + tn = − × dz t1 + t2 + ··· + tn t1, t2, . . . , tn n=1 t1+2t2+···+ntn=n n t +t +···+t ! Y 1 i i+1 n × zn i i=1   ∞ λ1
λ2
λn
λ1 ··· d X  X x λ2 λ3 0  n = −  λ λ λ  z dz  λ1 1 1 2 2 ··· n n  n=1 λ1+λ2+···+λn=n λ1>λ2>...>λn>0   ∞ λ1
λ2
λn
λ1 ··· X  X n · x λ2 λ3 0  n−1 = −  λ λ λ  z  λ1 1 1 2 2 ··· n n  n=1 λ1+λ2+···+λn=n λ1>λ2>...>λn>0 and d xez ln 1 − x(ez − 1)
= − dz 1 − x(ez − 1) ∞ ! ∞ ! X zn X zn = −x ω (x) n! n n! n=0 n=0 ∞ n ! X X ωk(x) = −x zn k!(n − k)! n=0 k=0 124 MIRCEA MERCA

∞ n ! X X n zn = −x ω (x) k k n! n=0 k=0 ∞ n−1 ! X X n zn = −x − x ω (x) + xω (x) k k n n! n=1 k=0 ∞ X zn (3.1) = −x − ω (x) + xω (x)
, n n n! n=1 where we have invoked (1.2). Thus we deduce that

λ1
λ2
λn
λ1 ··· (1 + x)ωn−1(x) X n · x λ2 λ3 0 = λ λ λ (n − 1)! λ1 1 1 2 2 ··· n n λ1+λ2+···+λn=n λ1>λ2>...>λn>0 and the theorem is proved.

4. Convolutions over weak compositions Recall that a composition of an integer n is a way of writing n as the sum of a sequence of positive integers. A weak composition of an integer n is similar to a composition of n, but allowing terms of the sequence to be zero. In this section, we show that the geometric polynomials ωn(x) and ωn+1(x) can be written as a summation over all weak compositions of n into exactly 3 parts.

Theorem 4.1. For n > 0, X  n  ω (x) = x ω (x)ω (x), n+1 a, b, c a b a+b+c=n where a, b, c are nonnegative integers. Proof. Using the Cauchy product of two power series, we obtain ∞ n ! ∞ n ! ∞ n−1 ! X z X z X ωn(x) X ωk(x) 1 − x ω (x) = 1 + − x zn n! n n! n! k!(n − k!) n=1 n=0 n=1 k=0 ∞ n−1 ! X X n zn = 1 + ω (x) − x ω (x) n k k n! n=1 k=0 = 1, where we have invoked (1.2). According to the proof of Theorem 1.2, we can write ∞ ∞ !−1 X zn d X zn x + (1 + x)ω (x) = ln 1 − x n n! dz n! n=1 n=1 ∞ ! d X zn = ln ω (x) dz n n! n=0 GEOMETRIC POLYNOMIALS 125

∞ ! ∞ !−1 X zn−1 X zn = ω (x) ω (x) n (n − 1)! n n! n=1 n=0 and ∞ ∞ ! ∞ ! X zn X zn X zn ω (x) = x + (1 + x)ω (x) ω (x) n+1 n! n n! n n! n=0 n=1 n=0 ∞ ! ∞ ! X zn X zn = −1 + (1 + x)ω (x) ω (x) n n! n n! n=0 n=0 ∞ n ∞ n ! X z X X ωk(x)ωn−k(x) = − ω (x) + (1 + x) zn. n n! k!(n − k)! n=0 n=0 k=0 Thus we deduce that n−1 ωn(x) ωn−1(x) X ωk(x)ωn−1−k(x) = − + (1 + x) (n − 1)! (n − 1)! k!(n − 1 − k)! k=0 or n nωn(x) ωn−1(x) X ωk−1(x)ωn−k(x) = − + (1 + x) n! (n − 1)! (k − 1)!(n − k)! k=1 or n nωn(x) ωn−1(x) X (1 + x)ωk−1(x) ωn−k(x) (4.1) = x + . n! (n − 1)! (k − 1)! (n − k)! k=2 On the other hand, by (3.1) we see that

n (1 + x)ωn−1(x) X x ωn−k(x)  , for n > 1, (4.2) = (n − 1)! (k − 1)! (n − k)! k=1 x, for n = 1. Taking into account (4.1) and (4.2), we obtain n k nωn(x) X X x ωk−j(x) ωn−k(x) = . n! (j − 1)! (k − j)! (n − k)! k=1 j=1 After a little manipulation we arrive at our identity.  For example, the weak compositions of 3 are: 0+0+3, 0+1+2, 0+2+1, 1+1+1, 0+3+0, 1+0+2, 2+0+1, 3+0+0, 1+2+0, 2+1+0. Thus the case n = 3 of Theorem 4.1 reads as follows:  3   3  ω (x) = x ω (x)ω (x) + 2 ω (x)ω (x) 4 0, 0, 3 0 0 0, 1, 2 0 1  3   3  + 2 ω (x)ω (x) + 2 ω (x)ω (x) 0, 2, 1 0 2 0, 3, 0 0 3 126 MIRCEA MERCA

 3   3   + ω (x)ω (x) + 2 ω (x)ω (x) 1, 1, 1 1 1 1, 2, 0 1 2 = x1 + 6x + 6(x + 2x2) + 2(x + 6x2 + 6x3) + 6x2 + 6x(x + 2x2)
= x + 14x2 + 36x3 + 24x4. Remark: Theorem 4.1 can be written in the following equivalent form: for n > 0, X  n  nω (x) = x aω (x)ω (x), n a, b, c b c a+b+c=n where a, b, c are nonnegative integers. As consequences of Theorem 4.1, we remark the following convolution identities involving geometric numbers and Genocchi numbers. These iden- tities seem to be new.

Corollary 4.3. For n > 0, X  n  ω = ω ω , n+1 a, b, c a b a+b+c=n where a, b, c are nonnegative integers. Corollary 4.4. For n > 1, 1 X  n  G = G G , n 2 − 2n a, b, c a b a+b+c=n where a, b, c are nonnegative integers. n Remark: Taking into account that Gn = 2(1 − 2 )Bn, the last corollary can be rewritten in terms of the Bernoulli numbers as follows: for n > 1, 1 X  n (1 − 2a)(1 − 2b) B = B B , n 1 − n a, b, c 1 − 2n a b a+b+c=n where a, b, c are nonnegative integers.

Acknowledgements The author likes to thank the referees for their helpful comments.

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Department of Mathematics, University of Craiova 200585 Craiova, Romania

Academy of Romanian Scientists, Ilfov 3, Sector 5, Bucuresti, Romania E-mail address: [email protected]