Combinatorial Unification of Binomial-Like Arrays

Transcription

1 University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School Combinatorial Unification of Binomial-Lie Arrays James Stephen Lindsay University of Tennessee - Knoxville Recommended Citation Lindsay, James Stephen, "Combinatorial Unification of Binomial-Lie Arrays. " PhD diss., University of Tennessee, This Dissertation is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact trace@ut.edu.

2 To the Graduate Council: I am submitting herewith a dissertation written by James Stephen Lindsay entitled "Combinatorial Unification of Binomial-Lie Arrays." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the reuirements for the degree of Doctor of Philosophy, with a major in Mathematics. We have read this dissertation and recommend its acceptance: Pavlos Tzermias, Xia Chen, Gina Pighetti (Original signatures are on file with official student records.) Carl G. Wagner, Major Professor Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School

3 To the Graduate Council: I am submitting herewith a dissertation written by James Stephen Lindsay entitled Combinatorial Unification of Binomial-Lie Arrays. I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the reuirements for the degree of Doctor of Philosophy, with a major in Mathematics. Carl G. Wagner Major Professor We have read this dissertation and recommend its acceptance: Pavlos Tzermias Xia Chen Gina Pighetti Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of Graduate Studies (Original signatures are on file with official student records.)

4 Combinatorial Unification of Binomial-Lie Arrays A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville James Stephen Lindsay May 2010

5 Copyright c 2010 by James S. Lindsay. All rights reserved. ii

6 Dedication I dedicate this thesis to Tammy and Steve, who started this ball rolling, to Cynthia and John, who iced it in the present direction, and to Heather, who has to live with it now. iii

7 Acnowledgments I would lie to express my sincerest gratitude to my advisor, Professor Carl G. Wagner, for helping me mae this so much more than I could have made it on my own. Thans are due as well to the remainder of the faculty and many of the graduate students in the Mathematics Department at the University of Tennessee, in particular my committee members: Pavlos Tzermias and Xia Chen. I also deeply appreciate my family for their enduring patience with me during this process. iv

8 Abstract This research endeavors to put a common combinatorial ground under several binomiallie arrays, including the binomial coefficients, -binomial coefficients, Stirling numbers, -Stirling numbers, cycle numbers, and Lah numbers, by employing symmetric polynomials and related words with specialized alphabets as well as a balls-and-urns counting approach. Using the method of statistical generating functions, - and p, -generalizations of the binomial coefficients, Stirling numbers, cycle numbers, and Lah numbers are all discussed as well, unified under a single general triangular array that is herein referred to as the array of Comtet-Lancaster numbers. v

9 Contents 1 Comtet s Theorem Notation Similarities Between Binomial Coefficients, Stirling Numbers, and Their - Analogues The Binomial Coefficients The Stirling Numbers of the Second Kind The -Binomial Coefficients The Carlitz -Stirling Numbers Comtet s Algebraic Unification A Combinatorial Interpretation of the Comtet Numbers Bijections to Familiar Structures and Combinatorial Proofs for the Special Cases The Binomial Coefficients The Stirling Numbers The -Binomial Coefficients The Carlitz -Stirling Numbers Additional Examples From Comtet s Note Lancaster s Theorem Similar Arrays Outside of Comtet s Unification The Cycle Numbers The Lah Numbers Lancaster s Algebraic Unification Combinatorial Interpretations of the Comtet-Lancaster Numbers Symmetric Polynomials Selections of Balls from Urns Bijections and Applications to Structures in the Special Cases The Cycle Numbers The Lah Numbers The Binomial Coefficients, Again The Comtet Case Revisited Comparison to a Similar Structure Additional Examples Available Via Statistical Generating Functions The -Binomial Coefficients A Variant on the -Binomial Coefficients vi

10 3.2 The Carlitz and p, -Stirling Numbers The and p, -Cycle Numbers The Comtet-Lancaster and p, -Lah Numbers Summary and Future Directions 73 Bibliography 76 Appendix 79 A Appendix: Partial Tables of Values of Arrays 80 Vita 93 vii

11 Chapter 1 Comtet s Theorem 1.0 Notation The notational conventions herein are as follows: N denotes the set {0, 1, 2,...} of nonnegative integers. P denotes the set {1, 2, 3,...} of positive integers. Q denotes the field of rational numbers. R denotes the field of real numbers. C denotes the field of complex numbers. F and F n, for n P, denote the finite field of elements, when = p d, for some prime p, and the n-dimensional vector space thereupon, respectively, with F 0 := {0}. For all n P, [n] := {1, 2,..., n}, with [0] :=. For all n N, [n] := {0, 1, 2,..., n}. Other more specialized notations appear as well 1. Some of those are as follows: For n P, and x any indeterminate or complex number, x n := x(x 1)(x 2) (x n + 1), with x 0 := 1, denotes the falling factorial function of degree n. For n P, and x as above, x n := x(x + 1)(x + 2) (x + n 1), with x 0 := 1, denotes the rising factorial function of degree n. 1 Some of these will have their formal definitions given later in the text, freuently in another (euivalent) form altogether. 1

12 For all n P, and an indeterminate, complex number, or, particularly, a power of a prime number, n := n 1 = n 1 1, with 0 := 0, denotes a -integer. For all n P, and p and indeterminates or complex numbers, including powers of prime numbers, n p, = p n 1 + p n 2 + p n p n 2 + n 1 = n p n p, with 0 p, := 0, denotes a p, -integer. For all n P and as above, n! := n (n 1) (n 2) 2 1, with 0! := 1 denotes the -factorial function, with p, -factorial defined analogously by 0 p,! := 1 and for all n P, For all n P, P, and as above, n p,! := n p, (n 1) p, 1 p,. n := n (n 1 )(n 2 ) (n ( 1) ) denotes the th -falling factorial, observing n = 0 for > n, in particular 0 = 0 for all P, and p, -falling factorials are defined analogously. 1.1 Similarities Between Binomial Coefficients, Stirling Numbers, and Their -Analogues The Binomial Coefficients Given n, N, let ( n ) := {A : A [n] and A = }, with N. These are the binomial coefficients. Immediately, it follows that Theorem For all n, N, whenever 0 n. ( ) n = n!!(n )! (1.1) Proof. Since for any n, N with n it is well nown (see, for instance, [20]) that n = n! (n )! counts the permutations of elements from [n], ( n ) counts the -element subsets of [n] by considering any permutation of -elements of [n] and mapping it to the (unordered) set of those elements, which is a!-to-one surjection. 2

13 and Theorem For all n, N, ( ) ( ) n n =. (1.2) n Proof. Straightforward by set complementation. Furthermore, two more results follow from the definition: First, Theorem Given N, for all n N, (x + 1) n = n =0 ( ) n x. (1.3) Proof. It more than suffices to establish this polynomial identity 2 for all r P. Then (1.3) taes the form n ( ) n (r + 1) n = r. (1.4) =0 Then both sides of (1.4) count the n-letter words in the alphabet [r]. The left-hand side does this by filling n slots with the r + 1 letters in [r]. The right-hand side does this in ( + 1 disjoint, exhaustive classes: those with exactly n zeros, for 0 n. The term n ) r chooses positions from among the n which will have elements of [r], and then the remaining n positions are all filled in with 0 s. By substitution of x 1 in place of x, (1.3) can be rewritten in the form x n = n =0 ( ) n (x 1). (1.5) Of course, (1.3) and (1.5) are special cases of the binomial theorem. And second, a two-term recurrence, Theorem For all n, P, ( ) n = ( ) n ( n 1 ), (1.6) subject to the boundary conditions ( ) 0 = δ0, for all N and ( n 0) = 1 for all n N. 2 A polynomial identity involving a polynomial of degree n reuires only n + 1 verified instantiations to establish the result. This fact is sometimes called the engineer s dream theorem and can be found and proved in [20], for instance. Here, the infinite number of instances in which this polynomial identity holds far exceeds the necessary n

14 Proof. The boundary conditions are obvious. For n, P, among all -element subsets of [n], ( n 1 1) counts those that contain the element n, and ( ) n 1 counts those that do not. From the recurrence (1.6), we derive the following formulas: First, a column generating function, Theorem For all N, n 0 ( ) n x n = x, (1.7) (1 x) +1 Proof. For every N, let g (x) := n 0 ( ) n x n. Then g 0 (x) = ( ) n 0 x n = x n = 1 1 x by the geometric series identity. Using the recurrence (1.6), a recurrence for g (x) arises for all P: g (x) = [( ) ( )] n 1 n 1 + x n 1 n 1 = ( ) n 1 x x n 1 + ( ) n 1 x x n 1 1 n 1 n 1 Whence, g (x) = = xg 1 (x) + xg (x). x 1 x g 1(x). The result follows by induction. And second, a closed-form expression, Theorem For all n, N, ( ) n = d 0 +d 1 + +d =n d i N 1, (1.8) Proof. The recurrence and boundary conditions in Theorem are recovered from the right-hand side of (1.8) by considering separately the cases in which d = 0 and otherwise. Alternatively, if choosing elements from the seuence (1, 2,..., n) to mae a -element subset of [n], then each d i counts the number of elements in the i th interval between the chosen elements of the seuence, taing the zeroth interval to mean the one preceding the first chosen element. Of course, all of these results are well nown, appearing, for instance, in [20]. 4

15 1.1.2 The Stirling Numbers of the Second Kind Given a fixed n N and for all N, let Π n, denote the set of partitions of [n] into nonempty blocs 3, and let S(n, ) := Π n,. These are the Stirling numbers of the second ind 4. Two results follow from this definition: First, recalling (1.5) and with the following proof appearing in [20], Theorem For any n N, x n = n S(n, )x. (1.9) =0 Proof. It more than suffices to establish this polynomial identity for all r P. Then (1.9) taes the form n r n = S(n, )r. (1.10) =0 Each side of (1.10) counts the functions f : [r] [n], the left-hand side directly. For the right-hand side, note that for each partition of [n] into nonempty blocs, there are!s(n, ) ordered partitions 5 of [n] into nonempty blocs since there are! possible orders for the blocs. Furthermore, by mapping any ordered partition of [n] into nonempty blocs to a function f : [n] [] given by f(i) = j whenever i [n] appears in bloc j [], notice that!s(n, ) counts the surjective functions f : [n] []. Thus, consider n S(n, )r = =0 n r!!s(n, )!(r )! = =0 n ( ) r!s(n, ). (1.11) For each [n], the term!s(n, ) ( r ) counts those functions f : [r] [n] in which range(f) = since ( r ) chooses the values in the range and!s(n, ) counts all surjective functions from [r] to that -element set. And second, a two-term recurrence similar to (1.6), Theorem For all n, P, =0 S(n, ) = S(n 1, 1) + S(n 1, ), (1.12) subject to the boundary conditions S(0, ) = δ 0,, for all N and S(n, 0) = 0, for all n P. Proof. The boundary conditions are straightforward, and when > n, Π n, =. Thus, assume n, P with 1 n. Then among those partitions of [n] into nonempty 3 A very common variant on this is distributions of n labeled balls into unlabeled urns so that no urn is left empty. 4 Unless otherwise indicated, the Stirling numbers will be taen to mean henceforth the Stirling numbers of the second ind unless explicit inclusion of the epithet is demanded for clarity or comparison, e.g. with the (signless) Stirling numbers of the first ind. 5 An ordered partition is one in which the order in which the blocs appear will distinguish one ordered partition from another. 5

16 blocs, S(n 1, 1) counts those in which the element n appears as the only element in its bloc, and S(n 1, ) counts those in which the element n appears in a bloc containing at least one other element, for which there are choices. From this recurrence, we derive the following formulas: First, a column generating function similar to (1.7), Theorem For all P, S(n, )x n = n 0 x (1 x)(1 2x) (1 x). (1.13) Proof. For every N, let g (x) := n 0 S(n, )x n. Then g 0 (x) = S(n, 0)x n = δ n,0 x n = 1. Using the recurrence (1.12), a recurrence for g (x) arises for all P: Whence, g (x) = g (x) = n 1 [S(n 1, 1) + S(n 1, )] x n = S(n 1, 1)x x n 1 + n 1 n 1 = xg 1 (x) + xg (x). x 1 x g 1(x). The result follows by induction. And second, a closed-form expression similar to (1.8), Theorem For all n, P, S(n, ) = d 0 + +d =n d i N S(n 1, )x x n 1 0 d 0 1 d1 d. (1.14) Proof. The recurrence and boundary conditions in (1.12) can be recovered from the righthand side of (1.14) by considering separately the cases in which d = 0 and otherwise. Another useful structure counted by S(n, ) is the set of restricted growth functions from [n] to [], first discovered by Stephen Milne [14]. Definition A surjective function f : [n] [] is a restricted growth function if in the seuence (f(1), f(2),..., f(n)), the first occurrence of j precedes the first occurrence of j + 1 for each j [ 1]. The set of restricted growth functions from [n] to [] will be denoted RGF(n, ). Before connecting restricted growth functions to partitions of a set, it is advantageous to designate a canonical form for writing such partitions. In particular, it is useful to 1. List the elements within the blocs in increasing order by their magnitudes, and 6

17 2. List the blocs in increasing order by the magnitudes of their smallest (here: initial) elements. Unless otherwise specified, henceforth such partitions are written in this canonical form. Theorem There is a bijection between the sets Π n, and RGF(n, ). Proof. Given a (canonically written) partition π Π n,, define a function f π : [n] [] by f π (i) = j whenever i appears in the j th bloc of π. Since the first occurrence of j will be from the smallest element of the j th bloc, and liewise for the first occurrence of j + 1, the canonical ordering on π provides that this map gives f π RGF(n, ). Furthermore, given any f RGF(n, ), by placing each i in bloc j of a partition of [n] with nonempty blocs whenever f(i) = j, it is clear that this map is surjective. Finally, since any two distinct members of Π n, have at least one element of [n] appearing in different blocs, their associated functions will return different values for such elements. All of these results are well nown, appearing, for instance, in [20] The -Binomial Coefficients Given a fixed n N and for all N and any power of a prime number, define the -binomial coefficient ( ) n to be the number of -dimensional linear subspaces of the n- dimensional vector space F n. This array is, in fact, a -analogue of the binomial coefficients, the latter being obtained from the former by choosing = 1 and letting [n] stand in place of F n 1, with subsets acting as the subspace-lie structure. A few results follow directly from this definition: First, -analogues of (1.1) and (1.5): Theorem For all n N and for a prime power, and x n = n =0 ( ) n = Proof. Formula (1.15) follows from the fact that n!!(n )!, (1.15) ( ) n (x 1)(x )(x 2 ) (x 1 ). (1.16) 1 n!!(n )! = ( n j ) ( j ) = n j=0 (1.17) are algebraic variants of the right-hand side of (1.15). The middle expression in (1.17) counts the -dimensional linear subspaces of F n by considering first the set C := {(x 1,..., x ) : the vectors x i are linearly independent in F n }. The map from C, which contains ( n 1)( n ) ( n 1 ) seuences, to the set of - dimensional subspaces of F n given by mapping a seuence of vectors to its linear span in F n is 7

18 a ( 1)( ) ( 1 )-to-one surjection since there are ( 1)( ) ( 1 ) ordered bases for each linear span, establishing (1.15). By performing a top-down summation and replacing ( n n ) with ( ) n, it follows that (1.16) can be proved by showing that for all r P, ( r ) n = n =0 ( ) n 1 n ( r i ). (1.18) Both sides of euation (1.18) count the linear transformations T : F n F r. The right-hand side does so since among all such transformations T, i=0 ( ) n 1 n ( r i ) i=0 counts those with a -dimensional null space. This can be verified by choosing first a - dimensional subspace W of F n and letting (x 1,..., x ) represent any ordered basis of W. Now, extend that basis to an ordered basis (x 1,..., x, x +1,..., x n ) of F n, noting that a linear transformation T : F n F r will have W as its null space precisely when T (x i ) = 0, for 1 i, and T (x j ), for + 1 j n, any linearly independent seuence of vectors in F r. These proofs appear in [21]. Observe that ( ) n 1 = n as a special case of (1.15). Furthermore, notice that by interchanging the roles of and n in (1.15), it follows directly that Theorem For all n, N, ( ) n = ( ) n. (1.19) n Additionally, there is a two-term recurrence analogous to (1.6) and similar to (1.12), Theorem For a prime power, with boundary conditions ( 0 ) = δ 0, and ( ) n 0 = 1 for every n, N, ( ) ( ) ( ) n n 1 n 1 = +, n, P. (1.20) 1 Proof. For > n, the recurrence holds in the form 0=0. Also, n, N, the boundary conditions are obvious. Thus, let n, P with n, and let W be any one-dimensional subspace of F n. Then among those -dimensional linear subspaces of F n, (i) ( ) n 1 counts those that contain W ) as a subspace, while (ii) ( n 1 counts those that do not. To see (i), let A be the set of linearly independent seuences (x 1,..., x ) in F n in which x 1 W. Note that A = ( 1)( n ) ( n 1 ) since there are ( 1) choices for x 1, ( n ) choices for a linearly independent x 2, and so on. Now, let B be the set of -dimensional linear subspaces of F n that also contain W as a subspace. Then the map 8 1

19 from (x 1,..., x ) to the linear span of (x 1,..., x ) is a ( 1)( ) ( 1 )-to-one surjection from A to B. Hence, B = ( 1)(n ) ( n 1 ) ( 1)( ) ( 1 ). The right-hand side of this can be simplified to ( n 1 1) ( n +1 1) ( 1 1) ( 1) = (n 1)! ( 1)!(n )! = ( ) n 1. 1 To see (ii), let A be the set of linearly independent seuences (x 1,..., x ) in F n in which no x i W. Note that A = ( n ) ( n ) since there are ( n ) choices for x 1 W, ( n 2 ) choices for a linearly independent x 2 W, and so on. Now, let B be the set of -dimensional linear subspaces of F n that do not contain W as a subspace. Then the map from (x 1,..., x ) to the linear span of (x 1,..., x ) is a ( ) ( 1 )-to-1 surjection from A to B. Hence, B = (n ) ( n ) ( ) ( 1 ). This gives ( B = (n 1)! n 1!(n 1)! = ). This proof also appears in [21]. From the recurrence (1.20), we derive other identities, each similar to a formula given for the binomial coefficients and Stirling numbers: First, a column generating function analogous to (1.7) and similar to (1.13), Theorem For every N and for a prime power, n 0 ( ) n x n = x (1 x)(1 x)(1 2 x) (1 x). (1.21) Proof. For every N, let g (x) := ( ) n x n. n 0 Then g 0 (x) = ( ) n 0 xn = x n = 1 1 x by the geometric series identity. Using the recurrence (1.20), a recurrence for g (x) arises for all P: g (x) = [ (n ) ( ) ] 1 n 1 + x n 1 n 1 = ( ) n 1 x 1 n 1 x n 1 + ( ) n 1 x x n 1 n 1 = xg 1 (x) + xg (x). 9

20 Whence, g (x) = x 1 x g 1(x). The result follows by induction. And second, a closed-form expression analogous to (1.8) and similar to (1.14), Theorem For every n, N and for an indeterminate, ( ) n = 0d 0+1d 1 +2d 2 + +d. (1.22) d 0 +d 1 + +d =n d i N Proof. The proof of this fact is similar to the proofs of Theorems and All of these results are well nown, appearing, for instance, in [21] The Carlitz -Stirling Numbers Following the approaches of Milne [14] and Wagner [19], given a fixed n N, let V be a -dimensional vector space over F. Then for each N, consider the set of seuences (U 1,..., U n ) of one-dimensional subspaces of V with dim(sp(u 1,..., U n )) =, where by Sp(U 1,..., U n ) is meant the linear span of the the spaces U 1,..., U. Associate with each such seuence a subseuence (U t1,..., U t ) obtained by letting U ti be the first instance in (U 1,..., U n ) in which dim(sp(u 1,..., U ti )) = i, for each i []. In [19], Wagner shows that the cardinality of the preimage of this map is the same for any choice of (U t1,..., U t ). Indeed he shows that, denoting the cardinality of the preimage of any such seuence by S (n, ), Theorem (Wagner). For every n, N and a power of a prime number, S (n, ) = d 1 + +d =n d i N (1 ) d 1 (2 ) d2 ( ) d. (1.23) These numbers S (n, ) are the Carlitz -Stirling numbers 6 (of the second ind), and two results follow from this definition: First, analogous to (1.9), Theorem For every n N and a power of a prime number, x n = n S (n, )x(x 1 )(x 2 ) (x ( 1) ). (1.24) =0 6 Carlitz first proposed these numbers in [2] in an investigation of a class of Abelian fields and subseuently expanded upon them in [3]. An interpretation similar to this one was discovered by Milne [14] by analyzing restricted growth functions. More generally, Wagner in [19] uses the idea of restricted growth on seuences of atoms in a modular binomial lattice of characteristic, resulting in a common treatment of S (n, ) for = 0 (chains), = 1 (sets), and a prime power (vector spaces). 10

21 Proof. It more than suffices to establish this polynomial identity for r for any r P with r n. Then (1.24) taes the form (r ) n = n S (n, )r (r 1 )(r 2 ) (r ( 1) ). (1.25) =0 Let n be fixed. Both sides of (1.25) enumerate the set A of seuences (U 1,..., U n ) of onedimensional linear subspaces of F r. That the left-hand-side does this is clear since there are r one-dimensional subspaces of F r and n positions in the seuence. For the right-hand-side of (1.25), let B denote the set of seuences of one-dimensional linear subspaces of F r with any length from 0 to n for which it holds that if the length of a seuence in B is, then dim(sp(u 1,..., U )) =. Now for each [n], let B denote the subset of B composed of seuences of length. Observe that (B 0,..., B n ) is an ordered partition of B and that for each [n], B = r (r 1 ) (r ( 1) ) since there are r choices for the first element of a seuence in B, r 1 choices for the second element of a seuence in B since it must be chosen outside of the span of the first choice, and so on. Further, by definition, if f maps A to B, then by definition any particular seuence in B has S (n, ) elements in its preimage under f, for each [n]. And second, analogous to (1.12), Theorem For all n, P and a power of a prime number, S (n, ) = S (n 1, 1) + S (n 1, ), (1.26) subject to the boundary conditions S (n, 0) = δ n,0, S (0, ) = δ 0,. Proof. The boundary conditions are clear. When n, P, consider the cases for which dim(sp(u 1,..., U n 1 )) = 1 and for which dim(sp(u 1,..., U n 1 )) =. In the first case, U t = U n. Since (U t1,..., U t 1 ) has preimage of cardinality S (n 1, 1) with elements of the form (U 1,..., U n 1 ), the preimage of the seuence (U t1,..., U t ) has elements of the form (U 1,..., U n ) and cardinality S (n 1, 1). On the other hand, when dim(sp(u 1,..., U n 1 )) =, U t U n. Therefore, we have U n Sp(U 1,..., U n 1 ). Since the elements of the preimage are of the form (U 1,..., U n 1 ), and since that set has cardinality S (n 1, ), given a seuence (U t1,..., U t ) it suffices to show that there are ways to choose U n. This, however, is precisely the number of one-dimensional subspaces of any -dimensional subspace over F, in particular of Sp(U 1,..., U n 1 ). From this recurrence, we derive a column generating function analogous to (1.13): Theorem For every N and a power of a prime number, S (n, )x n = n 0 x (1 x)(1 2 x)(1 3 x) (1 x). (1.27) Proof. For every N, let g (x) := n 0 S (n, )x n. 11

22 Then g 0 (x) = S (n, 0)x n = δ n,0 x n = 1. Using the recurrence (1.26), a recurrence for g (x) arises for all P: g (x) = [ ] S (n 1, 1) + S (n 1, ) x n n 1 Whence, g (x) = = S (n 1, 1)x x n 1 + n 1 n 1 = xg 1 (x) + xg (x). x 1 x g 1(x). The result follows by induction. These statements are all well nown, appearing in [8] for instance. S (n 1, ) x x n 1 Another structure counted by S (n, ), valid for any P, is presented in Section as a -analogue of RGF(n, ). 1.2 Comtet s Algebraic Unification There is strong similarity between the four primary euations provided in each of the four cases of the Section 1.1. In 1972, Louis Comtet drew them together with an algebraic unification in [4]. Theorem (Comtet s Theorem). Given a seuence b i i 0 in an integral domain I, the following four statements are euivalent specifications of a rectangular array C(n, ; b i ) for all n, N : 1. With boundary conditions C(0, ; b i ) = δ 0, and C(n, 0; b i ) = b n 0 for every n, N, C(n, ; b i ) = C(n 1, 1; b i ) + b C(n 1, ; b i ), n, P; (1.28) 2. For every n N, x n = n C(n, ; b i )ϕ (x), (1.29) =0 where ϕ 0 (x) 1, and ϕ (x) := (x b 0 )(x b 1 )(x b 2 ) (x b 1 ), P; 3. For every N, and C(n, ; b i )x n = n 0 4. For every n, N, C(n, ; b i ) = x (1 b 0 x)(1 b 1 x)(1 b 2 x) (1 b x) ; (1.30) d 0 +d 1 + +d =n d i N b d 0 0 bd 1 1 bd 2 2 bd. (1.31) 12

23 Proof. First, observe that each of the statements (1) through (4) uniuely defines a triangular array (C(n, ; b i )) n, N. Thus, it will suffice to show that those arrays defined by (2), (3), and (4) each satisfy the boundary conditions and recurrence given in (1). (2) (1): Direct examination of (1.29) provides that the boundary condition C(0, ; b i ) = δ 0, is satisfied. Furthermore, C(n, 0; b i ) = b n 0 is clear from the annihilation of the sum on the right-hand side of (1.29) that arises from the choice x = b 0. The recurrence is trivially obtained from (1.29) when > n, as both sides are 0. Thus, we may assume that 1 n. In this case, C(n, ; b i )ϕ (x) = x n = x x n 1 0 = x 0 C(n 1, ; b i )ϕ (x) = 0 C(n 1, ; b i )ϕ (x)(x b + b ) = 0 C(n 1, ; b i )(ϕ +1 (x) + b ϕ (x)) = 0(C(n 1, 1; b i ) + b C(n 1, ; b i ))ϕ (x). Since {ϕ (x)} 0 is a basis of the algebra I[x], the recurrence in (1.28) follows by comparison of coefficients. (3) (4): Given the column generating function in (1.30), applying the geometric series identity + 1 times provides b d 0 0 xd 0 b d 1 1 xd1 n 0 C(n, ; b i )x n = x d 0 0 = = n 0 d 1 0 each d i 0 Comparison of coefficients of x n produces (1.31). x n d 0 b d xd b d 0 0 bd 1 1 bd xd 0+d 1 + +d + d 0 + +d =n d i N b d 0 0 bd 1 1 bd. (4) (1): The boundary conditions of (1.28) are immediate from (1.31). When n, P, however, the sum in (1.31) is split into two cases: when d = 0 and when d > 0. Noting that n = (n 1) ( 1), this gives C(n, ; b i ) = d 0 + +d 1 =n d i N b d 0 0 bd 1 1 bd + d 0 + +d =n d i N;d P b d 0 0 bd 1 1 bd 13

24 = d 0 + +d 1 =n d i N b d 0 0 bd 1 1 bd + b d 0 + +d =n 1 d i N = C(n 1, 1; b i ) + b C(n 1, ; b i ) b d 0 0 bd 1 1 bd The values of the array C(n, ; b i ) are the Comtet numbers associated with b i i 0. Also, the function ϕ (x) defined in the second point above is called the th falling factorial function in b i i 0. Observe that Theorem reestablishes most of the results from Section 1.1 since from C(n, ; b i ),: 1. b i 1 gives the ordinary binomial coefficients, 2. b i = i, for all i N, gives the Stirling numbers, 3. b i = i, for all i N, gives the -binomial coefficients, and 4. b i = i, for all i N, gives the Carlitz -Stirling numbers. The statement and proof of Comtet s theorem as given here appear in [21] with little modification. 1.3 A Combinatorial Interpretation of the Comtet Numbers As suggested by Wagner in [23], let B i i 0 be a seuence of finite, pairwise disjoint sets, with B i = b i for each i N. Then, for all n, N, let W(n, ; b i ) denote the set of words of length n in B 0 B so that for all 0 i 1, every letter from the alphabet B i precedes the letters from the alphabet B i+1. The term from ascending alphabets will be used to mean that the letters in the words will be chosen in this manner. Theorem For all n, N and each seuence b i i 0 of nonnegative integers, W(n, ; b i ) = C(n, ; b i ). (1.32) Proof. For any word w W(n, ; b i ), for each i, if d i counts the number of letters in w chosen from the set B i, then for all n, N, W(n, ; b i ) = b d 0 0 bd 1 1 bd. (1.33) d 0 +d 1 + +d =n d i N Then the result follows by (4) in Comtet s Theorem In particular, C(n, 0; b i ) = b n 0 for all n N, and C(n, ; b i ) = 0 if 0 n <. Formula (1.33) may be rewritten in the form C(n, ; b i ) = 0 i 1 i 2 i n b i1 b i2 b in, (1.34) 14

25 by letting i j, for 1 j n, be the specific subscripts of the letters in a word w W(n, ; b i ). So we see that C(n, ; b i ) is the (n ) th complete symmetric function in b 0, b 1,..., b 7. Using (1.32), a combinatorial proof of (1.28) can be given when each b i N. Recall Euation (1.28): for all n, P, C(n, ) = C(n 1, 1) + b C(n 1, ), subject to the boundary conditions C(0, ) = δ 0, and C(n, 0) = b n 0, for all n, N. Proof. On the RHS of (1.28), C(n 1, 1) counts those words in W(n, ) that contain no letter from B, and b C(n 1, ) counts those that contain at least one letter from B. Before proceeding note that there is another pair of identities involving the Comtet numbers that do not appear above and yet can be proved via this interpretation, see [8]. These are two variants on analogues of the Hocey Stic Theorem. The first will be referred to as a diagonal variant while the second is called a vertical variant. Though they can both be proved simply by repeatedly expanding one of the terms in the recurrence, to prove them combinatorially it is helpful to introduce another perspective on the words in W(n, ; b i ). Notice that W(n, ; b i ) can also be represented as a union of the following sets: let B i denote the set of words in W(n, ; b i ) with the property that they contain at least one letter from the alphabet B i. Then clearly W(n, ; b i ) = B 0 B. A convenient way to express W(n, ; b i ) is to reorganize the above in terms of the pairwise disjoint sets B, B c B 1, B c Bc 1 B 2,..., B c Bc 1 B 0, where concatenation denotes the intersection of sets. Note that B c Bc j+1 B j, comprises those words in W(n, ; b i ) containing at least one letter from B j but no letters from any B i with i > j, for i, j [ 1], i.e. all of the words in W(n, ; b i ) with the property that the alphabet with the largest index among the alphabets B 0,..., B from which a letter appears is j. Thus, we claim that W(n, ; b i ) = (B ) (B c B 1) (B c Bc 1B 0 ), (1.35) i.e., the sets B, B c B 1, B c Bc 1 B 2,..., B c Bc 1 B 0 form a pairwise disjoint, exhaustive class of subsets of W(n, ; b i ). Proof. It remains to show that these sets are exhaustive within W(n, ; b i ). To do so, let w W(n, ; b i ). Then suppose that the last letter in w, i.e. the letter from the alphabet with largest index, is from B j. Then w B c Bc j+1 B j. Now first among the Comtet Hocey Stic Theorems is the diagonal variant: Theorem Subject to the same boundary conditions given in (1.28), n, P, C(n, ) = b j C(n 1 j, j). (1.36) j=0 7 Henceforth, occasionally the seuence b i i 0 will be omitted to condense the notation, though it will always be present when needed for clarity. 15

26 Proof. The term b j C(n 1 j, j) counts the words in B c Bc j+1 B j, and so (1.36) follows from (1.35). And second is the vertical variant, Theorem Subject to the same boundary conditions given in (1.28), n, P, C(n, ) = n j= b n j C(j 1, 1). (1.37) Proof. Among those words in W(n, ; b i ), the term b n j C(j 1, 1) counts those in which exactly n j letters, necessarily the last n j letters, are from the alphabet B. To see this, choose a word in W(n, ; b i ) end in n j letters from B. Map that word to the set W(j 1, 1; b i ) by deleting those n j letters from B. This map is a (b n j )-to-one surjection. 1.4 Bijections to Familiar Structures and Combinatorial Proofs for the Special Cases For the purpose of connecting W(n, ; b i ) with more familiar structures, we will use a canonical representation of the letters forming the alphabets B i. Specifically, represent each alphabet set B i (with B i = b i ), by B i = {b i,1, b i,2,..., b i,bi }. We adopt this convention to be able to determine the alphabet for each letter directly by inspection: the first subscript of each letter reveals to which alphabet it belongs The Binomial Coefficients When b i 1, the recurrence (1.28) taes the form C(n, ; 1 ) = C(n 1, 1; 1 ) + C(n 1, ; 1 ), for all n, P, (1.38) subject to the boundary conditions C(0, ; 1 ) = δ 0,, for all N, and C(n, 0; 1 ) 1. These are the same recurrence and boundary conditions satisfied by the binomial coefficients, given in (1.6), and so C(n, ; 1 ) = ( n ), so here the binomial coefficients enumerate the words in the set W(n, ; 1 ), i.e. words of length n in the ascending alphabets B 0 B, with each B i = {b i,1 }. Theorem For every n, N, there is a bijection between W(n, ; 1 ) and the set of -element subsets of [n]. Proof. In the cases when > n, the map is. When n = = 0, the map is from the empty word to the set containing. Now let n P and 0 n. Letting the j th letter of a word w W(n, ; 1 ) be w j = b ij,1, for j [n ], define a map from W(n, ; 1 ) to the -element subsets of [n] by w 1 w 2 w n {i 1 + 1, i 2 + 2,..., i n + n } c. (1.39) 16

27 That this map is a bijection is clear from construction, though it bears mentioning that since is the largest value i n can tae, and i 1 can eual 0, that the image of this set is indeed a -element subset of [n]. As an aside, there is a second interpretation available from these words, yielding another structure counted by the binomial coefficients essentially for free. Collecting, in order, the seuence of initial subscripts of a word in W(n, ; 1 ) provides the image of a function f : [n ] [] that is monotonically increasing. Thus, the binomial coefficients count the number of such functions. Also, let w W(n, ; 1 ). Then if one taes account of d i, the number of letters in w with initial subscript i, as i ranges from 0 to, then d 1 + d d = n, and there is only one possible manifestation of w. For all n, N, this yields the formula ( ) n = d 1 +d 2 + +d =n d i N 1. (1.40) Furthermore, if the specific initial subscripts in w are 0 i 1 i 2 i n, then there is still only one possible manifestation of w, providing the similar formula ( ) n = 1, (1.41) also valid for all n, N. 0 i 1 i 2 i n Finally, the two variants on the Hocey Stic Theorem given in Theorems and tae the forms, for all n, P, and ( ) n = ( ) n 1 j, (1.42) j j=0 ( ) n = n j= both subject to the same boundary conditions given in (1.6). ( ) j 1, (1.43) 1 In light of the proofs of Theorems and and the bijection (1.39) above, among all -element subsets of [n], the term ( ) n 1 j j in (1.42) counts those in which all of the j elements of [n] larger than n j are present but n j is not, i.e. n j is the largest excluded element, and the term ( j 1 1) in (1.43) counts those in which the largest element of [n] present is j. These interpretations are in agreement with the ones available by direction inspection of (1.42) and (1.43) using the provided interpretation of the binomial coefficients. 17

28 1.4.2 The Stirling Numbers When b i = i, for all i N, the recurrence (1.28) taes the form C(n, ) = C(n 1, 1) + C(n 1, ), for all n, P, (1.44) subject to the boundary conditions C(0, ) = δ 0,, for all N, and C(n, 0) = 0, for all n P. These are the same recurrence and boundary conditions satisfied by the Stirling numbers, given in (1.12), and so C(n, ; i ) = S(n, ). Thus, the Stirling numbers also enumerate the words in W(n, ; i ), i.e. the words of length n in the ascending alphabets B i := {b i,1, b i,2,..., b i,i }, for i [], taing note that B 0 =. Recall that RGF(n, ) is the set of restricted growth functions f : [n] [], i.e. surjections f that are surjections with the property that the first occurrence of j precedes the first occurrence of j + 1, for j [ 1], in the seuence (f(1),..., f(n)). Theorem There is a bijection between W(n, ; i ) and RGF(n, ) for every n, N. Proof. When n = 0, = 0, or > n with n, P, the map is in every case except when n = = 0, in which case the empty word maps to the trivial restricted growth function that can be represented by the empty seuence. For n, P, with 1 n, consider the map from W(n, ; i ) to RGF(n, ) defined for w W(n, ; i ) by first inserting the different values of [] in increasing order into w so that all of the heretofore unplaced elements of [] up to j together with j are placed immediately before the first occurrence of some letter in w from the alphabet B j. All remaining elements of [] are placed at the end of this expanded word, also in increasing order. Then map the resulting n-letter word to a seuence of length n by taing the inserted elements to themselves and each letter b j,ij of w to i j, preserving the order in which they appear. This seuence is clearly uniue to w and can be understood as the image of a function f RGF(n, ). A concrete example of this map is helpful for clarity: Example Let n = 11 and = 5. Consider the word Then w maps first to w = b 1,1 b 3,2 b 3,1 b 3,1 b 4,3 b 4,2 W(11, 5; i ). (1.45) 1 b 1,1 2 3 b 3,2 b 3,1 b 3,1 4 b 4,3 b 4,2 5, (1.46) which is identified with the restricted growth function from the set [11] to the set [5] (1, 1, 2, 3, 2, 1, 1, 4, 3, 2, 5). (1.47) Observe also that if for a word w W(n, ; i ), one taes account of d i, the number of letters in w with initial subscript i, as i ranges from 1 to, then d 1 + d d = n, and there are 1 d 1 2 d2 d possible manifestations of w. For all n, N, this yields the formula S(n, ) = 0 d 0 1 d1 d. (1.48) d 0 +d 1 + +d =n d i N 18

29 Furthermore, if the specific initial subscripts in w are 0 i 1 i 2 i n, then there are i 1 i 2 i n possible manifestations of w, providing the similar formula S(n, ) = i 1 i 2 i n, (1.49) also valid for all n, N. 0 i 1 i 2 i n Finally, the two variants on the Hocey Stic Theorem given in Theorems and tae the forms, for all n, P: and S(n, ) = ( j)s(n 1 j, j), (1.50) j=0 S(n, ) = n n j S(j 1, 1), (1.51) j= both subject to the same boundary conditions as (1.12). In light of the proofs of Theorems and and the bijection described in Theorem 1.4.2, among all elements of RGF(n, ), the term ( j)s(n 1 j, j) in (1.50) counts those in which the last j values in the image seuence are f(n j +1) = j +1,..., f(n) =, with either f(n j) < j or f(n j 1) = f(n j) = j, and the term n j S(j 1, 1) in (1.51) counts those in which the first occurrence of f(i) = in the image seuence is at i = j. These interpretations are in agreement with the ones available by direction inspection of (1.50) and (1.51) using the provided interpretation of the Stirling numbers in terms of restricted growth functions The -Binomial Coefficients When b i = i, i N, the recurrence (1.28) taes the form C(n, ) = C(n 1, 1) + C(n 1, ), for all n, P, (1.52) subject to the boundary conditions C(0, ) = δ 0,, for all N, and C(n, 0) 1. These are the same recurrence and boundary conditions satisfied by the -binomial coefficients, given in (1.20), and so C(n, ; i ) = ( ) n. Thus, here the -binomial coefficients count the words in W(n, ; i ), i.e. words of length n in the ascending alphabets B 0 B with each B i := {b i,1, b i,2,..., b i, i}. This interpretation is valid for every P and can be extended to every N by taing B i = when = 0 and i 0. 19

30 A bijection between these words and the set of -dimensional linear subspaces of F n would be desirable. One can be forced, though it is unsatisfactory, by considering the vector spaces in terms of n echelon matrices with entries in F with all rows nonzero and using the lexicographical order. Note that for w W(n, ; i ), if one taes account of d i, the number of letters in w with initial subscript i, as i ranges from 0 to, then d 1 +d 2 + +d = n, then there are 0d 0+1d 1 +2d 2 + +d possible manifestations of w. For all n, N, this yields the formula ( ) n = 0d 0+1d 1 +2d 2 + +d. (1.53) d 1 +d 2 + +d =n d i N Furthermore, if the specific initial subscripts in w are 0 i 1 i 2 i n, then there are i 1+i 2 + +i n possible manifestations of w, providing the similar formula ( ) n = i 1+i 2 + +i n, (1.54) also valid for all n, N. 0 i 1 i 2 i n In addition, the two variants on the Hocey Stic Theorem given in (1.3.2) and (1.3.3) tae the forms, for all n, P: and ( ) n = ( ) n = j=0 n j= ( ) n 1 j j, (1.55) j ( ) j 1 (n j), (1.56) 1 both subject to the same boundary conditions as (1.20). By the symmetry of ( ) n, there is another variant of each -binomial Hocey Stic Theorem: Theorem For all n, N and an indeterminate, and ( ) n = ( ) n = j=0 n j= subject to the same boundary conditions as (1.20). ( ) j 1 j, (1.57) 1 ( ) n 1 j j(n ), (1.58) j Proof. To see (1.57), consider (1.55) and apply the symmetry of the -binomial coefficient 20

31 along with a top-down summation: j=0 ( ) n 1 j j j = = j=0 n j=n ( ) n 1 j j n 1 j (n ) ( j 1 n 1 ). Interchanging the roles of and n yields (1.57). To see (1.58), consider (1.56) and apply the symmetry of the -binomial coefficient along with a top-down summation: n j= ( ) j 1 (n j) 1 = = n j= n j=0 ( ) j 1 (n j) j ) j ( n 1 j n j Interchanging the roles of and n yields (1.58). These are not explained by (1.3.2) and (1.3.3). In fact, they are special to ( ) n and do not apply to C(n, ; b i ) in general since C(n, ; b i ) is not symmetric in and n for an arbitrary seuence b i i The Carlitz -Stirling Numbers When b n = n, for all n N, the recurrence (1.28) taes the form C(n, ) = C(n 1, 1) + C(n 1, ), for all n, P, (1.59) subject to the boundary conditions C(0, ) = δ 0,, for all N, and C(n, 0) = 0, for all n P. These are the same recurrence and boundary conditions satisfied by the Carlitz -Stirling numbers, and so C(n, ; i ) = S (n, ). Thus, S (n, ) counts the words in W(n, ; i ), i.e. the words of length n the ascending alphabets B 1 B with each B i := {b i,1, b i,2,..., b i,i }, taing note that B 0 = need not be included. This is valid for any P. Since the words in W(n, ; i ), counted by S(n, ) have a connection to restricted growth functions, it would be nice if the words in the -analogous W(n, ; i ) do as well. Such a connection can be had in a new way by extending the notion of a restricted growth function in a way that slightly relaxes the usual notion of restricted growth, which will apply whenever P. For notational convenience, the notation [j ] := [j ] [(j 1) ], will be applied. For n, N and P, let { RGF(n, ; ) := f : [n] [ ] : the first occurrence of an element in 21

32 [j ] precedes the first occurrence of an element in [(j + 1) ], } j [], with at least one element from each [j ] present be the set of -analogized restricted growth functions, more conveniently referred to as restricted growth functions 8. Refer to those restricted growth functions in which the first appearance of an element belonging to each [j ] is the largest possible among those, i.e. j, by canonical -restricted growth functions. Denote the set of these by RGF (n, ; ). Remar For all n, N and any P, RGF(n, ; ) = ( 2) RGF (n, ; ). (1.60) Proof. This follows since there are j 1 possible choices for a first element belonging to [j ] for each j []. Hence, Theorem For every n, P, RGF (n, ; ) = S (n, ). (1.61) Proof. The method of proof will be to show that there is a bijection from the words in W(n, ; i ) to RGF (n, ; ). Observe first that only the subscripts of the letters of the words matter, as they are all b s. In fact, given such a word, it can be represented instead as a seuence of ordered pairs of the form ((1, i 1,1 ), (1, i 1,2 ),..., (1, i 1,d1 ), (2, i 2,1 ), (2, i 2,2 ),..., (2, i 2,d2 ),... (, i,1 ), (, i,2 ),..., (, i,d )), (1.62) where each i l,m [l ]. Now, into that seuence, insert j along with i for every i < j not yet inserted immediately before the first instance of an ordered pair with first term j, for each j []. If the largest first term is less than, then the remaining numbers of the form i should be placed in increasing order at the end. Since the seuence (1.62) is of length n, and values have been added, the result is a seuence of length n. Finally, map the resulting seuence to a canonical restricted growth function, here represented by a seuence (s 1, s 2,..., s n ) in which those items that are not ordered pairs are mapped to themselves and those that are get mapped to their second components. Corollary By Remar and Theorem 1.4.6, for all n, N and P, RGF(n, ; ) = ( 2) S (n, ). (1.63) The numbers ( 2) S (n, ) are often denoted by S (n, ). In this wor they have the interpretation of counting the restricted growth functions from [n] to [ ]. Further discussion 8 Note that one must tae care with the term restriction and realize it is applied for brevity 22

33 of S (n, ) is omitted since they are not Comtet numbers 910. Observe that for w W(n,, i ), if one taes account of d i, the number of letters in w with initial subscript i, as i ranges from 1 to, then d 1 +d 2 + +d = n, and there are (1 ) d 1 (2 ) d2 ( ) d possible manifestations of w. For all n, N, this yields the formula S (n, ) = d 1 +d 2 + +d =n d i N (1 ) d 1 (2 ) d2 ( ) d. (1.64) Furthermore, if the specific initial subscripts in w are 0 i 1 i 2 i n, then there are (i 1 ) (i 2 ) (i n ) possible manifestations of w, providing the similar formula also valid for all n, N. S (n, ) = 0 i 1 i 2 i n (i 1 ) (i 2 ) (i n ), (1.65) Finally, the two variants on the Hocey Stic Theorem given in (1.3.2) and (1.3.3) tae the forms, for all n, P: and S (n, ) = S (n, ) = ( j) S (n 1 j, j), (1.66) j=0 n ( ) n j S (j 1, 1), (1.67) j= both subject to the same boundary conditions as (1.26). In light of the proofs of Theorems and and the bijection in the proof of Theorem 1.4.6, among all elements of RGF(n, ), the term ( j) S (n 1 j, j) in (1.66) counts those in which the last j values in the image seuence are f(n j + 1) = ( j) + 1,..., f(n) =, with either f(n j) < ( j) or f(n j 1) = f(n j) = ( j), and the term ( ) n j S (j 1, 1) in (1.67) counts those in which the first occurrence of f(i) = ( 1) in the image seuence is at i = j. Observe that these interpretations could be arrived at by direct inspection of (1.66) and (1.67). 9 For further reading on the subject, see [14] or [19] 10 The above construction and proofs apply to a class of restricted growth functions counted by the fully general Comtet numbers by replacing each j with b j, mutatis mutandis. 23

34 1.5 Additional Examples From Comtet s Note In Comtet s paper [4], he gives a series of examples of what are now called Comtet numbers on the first page, most with combinatorial interpretations. Two such (related) examples are, in the present notation, C(n, ; (2i) 2 ) and C(n, ; (2i + 1) 2 ), which he states count the partitions of [2n] into 2 blocs (respectively of [2n + 1] into blocs) where the bloc cardinality of each bloc is odd. Since he does not offer proofs of these statements, the goal of this section will be to give such verification explicitly. Our notations for these sets will be Π (1) 2n,2 and Π(1) 2n+1,2+1, respectively. To see Comtet s claim, what needs to be established is that the given structures satisfy the recurrences and boundary conditions of C(n, ; (2i) 2 ) and C(n, ; (2i + 1) 2 ). Specifically, and C(n, ; (2i) 2 ) = C(n 1, 1; (2i) 2 ) + (2) 2 C(n 1, ; (2i) 2 ), (1.68) C(n, ; (2i + 1) 2 ) = C(n 1, 1; (2i + 1) 2 ) + (2 + 1) 2 C(n 1, ; (2i + 1) 2 ), (1.69) both subject to the boundary conditions C(n, 0; ) = δ n,0 and C(0, ; ) = δ 0,. Proof. Since the two separate cases given in (1.68) and (1.69) are proved similarly, mutatis mutandis, only the case in which b i = (2i) 2 will be proved explicitly. Furthermore, the boundary conditions are obvious, following just as in the case for the Stirling numbers. Among those partitions π Π (1) when 2n,2, consider separately the disjoint, exhaustive cases 1. both of the elements 2n 1 and 2n of [2n] appear in blocs of cardinality 1; 2. the elements 2n 1 and 2n occupy the same bloc; and 3. the elements 2n 1 and 2n occupy distinct blocs not both of which are singletons. Given a partition in Π (1) 2n,2 described in the first of these cases, consider the bijection to Π (1) 2n 2,2 2 given by deleting from the end of that partition the two singleton blocs containing the elements 2n 1 and 2n. Thus, there are C(n 1, 1; (2i) 2 ) partitions of [2n] into 2 blocs, each with odd cardinality, in which the elements 2n 1 and 2n both appear in a bloc with cardinality 1. Given a partition in Π (1) 2n,2 described in the second of these cases, consider the map to Π (1) 2n 2,2 given by deleting the elements 2n 1 and 2n from whichever of the 2 blocs that they appear together in, resulting in a bloc with odd cardinality. Given a partition in Π (1) 2n,2 described in the third case, consider the map to Π(1) 2n 2,2 given by deleting the elements 2n 1 and 2n from whichever distinct two of the 2 blocs that they appear in, which results in those two blocs each having even, nonzero cardinalities. In that case, examine the two blocs that contained 2n 1 and 2n, and among those two blocs, move the smallest element among them to the other bloc so that both have odd cardinality. The map that considers both of the second and third cases together is a 24