G. C. Rota, "On the foundations of combinatorial theory," Zeitschrift Fur Wahrscheinlichkeitstheorie Und Verwandte Gebietee, vol. 2, pp. 340--368, 1964.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... M as a formal series of ZhhMii and call it the M obius series of M . A one line computation shows that M is the formal inverse of the characteristic series: m) M = M ( m) 1 : 1) This identity is called a M obius inversion formula, see Cartier and Foata [3] Lallement [15] or Rota [16] in a di erent setting. The classical M obius inversion principle in number theory (see [13] Chapter XVI) is a special instance of the identity. However, it is generally dicult, given a monoid with the nite decomposition property, to e ectively compute the M obius series. It is easily checked ....

G.-C. Rota. On the foundations of combinatorial theory. I. Theory of Mobius functions. Z. Wahrscheinlichkeitstheor. Verw. Geb., 2:340-368, 1964.

....can enumerate all simplices of and compute the Euler characteristic in polynomial time. Currently the fastest way to compute the Euler characteristic is to determine V = fS : S is an intersection of facets of g and then compute ( in time O jV j by a M obius function approach, see Rota [54]. Usually V is much smaller than the whole face lattice of . V can be listed lexicographically by an algorithm of Ganter [23] in time O(minfm;ng jV j) where is the number of vertex facets incidences. 32. f Vector of Simplicial Complexes Output: The f vector of Status (general) ....

G.-C. Rota, On the foundations of combinatorial theory { I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie, 2 (1964), pp. 340-368.

....We map each into v , where (i) the entries of v are indexed by subsets of E n , and (ii) For S . n , v S = 1 i# v j = 1 for all j S. Clearly, for 1 n, v = v j , so each v is mapped into a distinct column of the zeta matrix of the subset lattice L of E n (see [R64]) For simplicity, we will forgo the standard lattice theoretic notation (#, #, #) and use the corresponding set theoretic operators instead (#, and identify elements of the lattice with subsets of E n . For completeness, we state the definition of Z: it has a row and a column for each ....

G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrsch. Verw. Gebiete 2 (1964), 340-368.

....otherwise B is dependent. If B is independent then we say that B is a base for xB . If B is a minimal (with respect to inclusion) dependent set then we say that B is a circuit. Given a total order on A, then a broken circuit is obtained by taking a circuit and removing its smallest atom. Rota [16] first stated an important theorem giving an interpretation to the Mobius function in terms of broken circuits. He did this for geometric lattices, i.e. those 1 which are semimodular and where every element is a join of atoms. It is not hard to generalize this result to lattices which are just ....

G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie, 2 (1964), 340--368.

....a Hopf algebra structure. That is, there is an endofunction of the bialgebra, called the antipode, that fulfills a certain defining relation. One can view the antipode as a generalization of the Mo# bius function, which has played an important role in the theory of posets since Rota s seminal work [9]. Moreover, Schmitt gives a closed formula for the antipode which is a generalization of Philip Hall s formula for the Mo# bius function. In Sections 8 and 9 we extend the reduced incidence Hopf algebra of posets to a Hopf algebra of hierarchical simplicial complexes. To construct the reduced ....

G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Mo# bius functions, Z. Wahrsch. Verw. Gebiete 2 (1964), 340#368.

....the integers. We also give some applications of this second main theorem, including the Tamari lattices. 1 Bounded below sets In a fundamental paper [25] Whitney showed how broken circuits could be used to compute the coefficients of the chromatic polynomial of a graph. In another seminal paper [20], Rota refined and extended Whitney s theorem to give a characterization of the Mobius function of a geometric lattice. Then one of us [21] generalized Rota s result to a larger class of lattices. In this paper we will present a theorem for an arbitrary finite lattice that includes all the others ....

.... Theta be any total order on A(L) Then each circuit C gives rise to a broken circuit C = C n c where c is the first element of C under Theta. A set B A(L) is NBC (no broken circuit) if B does not contain any broken circuit and in this case B is an NBC base for x = B. Rota s NBC theorem [20] is as follows. Theorem 1.2 (Rota) Let L be a finite geometric lattice and let Theta be any total order on A(L) Then for all x 2 L we have = Gamma1) ae(x) Delta (number of NBC bases of x) To derive this result from Theorem 1.1, we first prove that when L is geometric and Theta is ....

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G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340--368.

....an arrangement A, let (X) 0; X) denote the Mobius function of the lattice L(A) it is uniquely defined by Y X (Y ) ffi 0;X where ffi 0;X is the Kronecker delta. The Mobius function is one of the fundamental invariants of any partially ordered set; see the seminal article of Rota [16]. The characteristic polynomial of A is : 2) Since the characteristic polynomial is just the generating function for the Mobius function, it is also of prime importance. Our results in this paper give a combinatorial interpretation for the characteristic polynomials of hyperplane ....

....Recall that L(A) is ordered by reverse inclusion so that Y ae X. In particular g(R ) j[ Gammas; s] Aj. Note also that X [ Gammas; s] is combinatorially just a cube of dimension dimX and side t so that f(X) t . Finally, f(X) Y X g(Y ) so by the Mobius Inversion Theorem [16] j[ Gammas; s] Aj = g( 0) X)f(X) A; t) which is the desired result. In the proof of Theorem 2.1, it was crucial that each of the subspaces X under consideration had exactly t dim(X) points in [ Gammas; s] In fact, the only subspaces of R with this property are those ....

G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340--368.

....may be useful. The theorem asserts that, if G is a transitive extension of H, and A is an integral domain, then so is (and the same holds for the stronger condition of the conjecture) 2 The theorem be a set, usually assumed to be countably infinite. The reduced incidence algebra (Rota [8]) A of the poset of finite subsets of Omega is defined as follows. For n 0, let V n be the vector space (over C ) of functions from the set n of n element subsets of Omega to C . Addition and scalar multiplication of functions is defined pointwise. We define a multiplication from V k ....

G.-C. Rota, On the foundations of combinatorial theory, I: Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340--368.

.... Phi Phi Phi Phi r r r r r r r r a ba ab e aba bab b w o 1 2 4 1 2 3 4 4 1 4 1 Figure 5. Tilted Bruhat order D a (W ) for W of type B 2 Recall [21] that a finite graded poset with 0 and 1 (resp. with 0) is called Eulerian (resp. lower Eulerian) if its Mobius function [20] is given by (x; y) Gamma1) for any x y. A well known (but non trivial cf. 2, 8, 18] theorem of Verma [23, 24] asserts that any interval in the Bruhat order of any Coxeter group is Eulerian. To our knowledge, no simple proof of this result is known, except for the special case x = e ....

G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, in: Gian-Carlo Rota on combinatorics, Birkhauser, 1995, pp. 3-31.

.... The divisor poset D 18 Figure 1: Some example posets One of the advantages of the combinatorial Mobius function is that its inversion theorem unifies and generalizes the previous three results. In addition, it makes the definition (2) transparent, encodes topological information about posets [6, 40], and has even been used to bound the running time of certain algorithms [9] We will now define this powerful invariant. Let finite P be a poset with partial order #.IfP has a unique minimal element then it will be denoted 0= 0 P , and if it has a unique maximal element then we will use the ....

.... (so called because in I(D n ) it is related to the Riemann zeta function) given by #(x, y) 1 for all intervals [x, y] It is easy to see that # is invertible in I(P)andinfactthat# 1 = where is defined by (4) The fundamental result about is the combinatorial Mobius Inversion Theorem [40]. Theorem 2.4 Let P be a finite poset and f,g : P # C. y,x)f(y) 4 (x, y)f(y) It is now easy to obtain the Theorems 2.1, 2.2, and 2.3 as corollaries by using Mobius inversion over D n , B n ,andC n , respectively. For example, to get the Principle of Inclusion Exclusion, use f,g : B n ....

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G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340--368.

....that is, # # : Clearly, # # is a lattice with respect to the usual inclusion order. The following interpretation of dom # (X)intermsoftheMobius function of # # is due to Manthei [Man90, Man91] and proved here in a new and simplified way without making use of Rota s crosscut theorem [Rot64]. Let us first define the Mobius function: Definition 5.1.13 The Mobius function of a finite partially ordered set P with least element 0 is the unique # valued function P on P such that for any x (5.3) P (y) # 0x where # is the usual Kronecker delta. Proposition 5.1.14 [Man90, ....

....are drawn for the Tutte polynomial, the characteristic polynomial and the # invariant of a matroid, the Euler characteristic of an abstract simplicial complex and the Mobius function of a partially ordered set. In particular, we rediscover a recent generalization of Rota s crosscut theorem [Rot64] due to Blass and Sagan [BS97] and obtain a new proof of a classical theorem due to Weisner [Wei35] A key role in proving these results is due to Theorem 3.2.4 and its forthcoming generalization to partially ordered sets. 6.1 Inclusion exclusion on partition lattices In this section, we ....

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G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 2 (1964), 340--368.

....compact support. 4.2. The umbral representation. A sequence (a n ) n=0 is said to be unital if a 0 = 1. The moment sequence of a random variable is such an example. For unital sequences, umbral calculus is a powerful working tool. We refer to Roman and Rota [12] and Rota, Kahaner and Odlyzko [13] for detailed exploration on umbral calculus. Most recent development can be found in Rota and Taylor [16, 17] Rota, Shen and Taylor [15] and Rota and Shen [14] Umbrae are usually denoted by Greek letters ; Associated to an umbra is in fact a linear functional L on the formal ....

G.-C. Rota, D. Kahaner, and A. Odlyzko. On the foundations of combinatorial theory. VIII. nite operator calculus. J. Math. Anal. Appl., 42:684-760, 1973.

....is, F # : Clearly, F # is a lattice with respect to the usual inclusion order. The following interpretation of dom F (X) in terms of the Mobius function of F # is due to Manthei [Man90, Man91] and proved here in a new and simplified way without making use of Rota s crosscut theorem [Rot64]. Let us first define the Mobius function: Definition 5.1.13 The Mobius function of a finite partially ordered set P with least element 0 is the unique Z valued function P on P such that for any x (5.3) P (y) # 0x , where # is the usual Kronecker delta. Proposition 5.1.14 [Man90, ....

....are drawn for the Tutte polynomial, the characteristic polynomial and the # invariant of a matroid, the Euler characteristic of an abstract simplicial complex and the Mobius function of a partially ordered set. In particular, we rediscover a recent generalization of Rota s crosscut theorem [Rot64] due to Blass and Sagan [BS97] and obtain a new proof of a classical theorem due to Weisner [Wei35] A key role in proving these results is due to Theorem 3.2.4 and its forthcoming generalization to partially ordered sets. 6.1 Inclusion exclusion on partition lattices In this section, we ....

[Article contains additional citation context not shown here]

G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 2 (1964), 340--368.

....P ThetaP Z defined inductively by (a; b) 1 if a = b, Gamma ac b (a; c) else. This can be rewritten in the useful and more intuitive form acb (a; c) ffi a;b (6) where ffi a;b is the Kronecker delta. For more information about Mobius functions, see the seminal article of Rota [23] or the book of Stanley [28] The Mobius function of Pi n is well known. In particular 0; 1) Gamma1) n Gamma1 (n Gamma 1) where 1 = 12 : n is the unique maximal element of Pi n . This is enough to determine on any interval of this lattice. For example, for any = B 1 =B 2 = ....

....weakly decreasing order) Of course, 0; is just the type of . All of the rest of the proofs in this section will be based on the Mobius Inversion Theorem. This result was first proved in somewhat less generality by Weisner [33] The reader is encouraged to consult Rota s influential article [23] for more details. Theorem 3.2 (Mobius Inversion Theorem) Let P be a poset, let (G; be an abelian group, and consider two functions f; g : P G. Then (a; b)f(b) for all a 2 P : Dually (b; a)f(b) for all a 2 P : We will also need a simple corollary of this theorem that slightly ....

G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340--368.

....neighbourhood; e) clique types for nearest neighbour neighbourhood; f) additional clique types for second order neighbourhood. A. A Representation for Potential V The following construction for the Potential is due to Grimmett [25] which first requires the Mobius inversion theorem [41]. Theorem 1: Mobius inversion theorem for arbitrary real functions F and G defined on the subsets B and C of some finite set A: G(B) i# G(B) 1) B C F (C) 4) or, equivalently, 1) B C F (C) 5) A = number of sites in set A. Moussouris [36] developed an elegant proof ....

G. C. Rota, "On the foundations of combinatorial theory," Zeitschrift Fur Wahrscheinlichkeitstheorie Und Verwandte Gebietee, vol. 2, pp. 340--368, 1964.

....(3) where I s rl is the Euclidean distance between two points s,r E , and o is the order of the neighbourhood system, see Fig. 1) A. A Representation for Af Potential V The following construction for the Af Potential is due to Grimmeft [25] which first requires the MSbius inversion theorem [43]. Theorem 1: MSbius inversion theorem for arbitrary real functions F and G defined on the subsets B and C of some finite set A: or, equivalently, G(B) E ( 1)l l lCIF(C) CCB (4) F(A) E E ( 1)I I IclF(C) BCA C CB (5) where IAI = number of sites in set A. Moussouris [38] developed an ....

G. C. Rots, "On the foundations of combinatorial theory," Zeitschri/.t Fur Wahrschein- lichkeitstheorie Und Verwandte Gebietee, 2, pp. 340-368, 1964.

....for all finite sets C, ij)aeC u Gamma B jCj; 6:23) then, for any finite set X, a) fi fi fi ds jfi u s (ij)j fijXjB t0 (6.24) b) fi fi fi ds jfi u s (ij)je ; 6.25) where B ts j B d . Remarks: Inequality (6. 22) was the first one proved [55, 48, 52]. It can be more dramatically stated in the form fi fi fi fi fi G fi fi fi fi Gamma 1j ; 6:26) which shows the large amount of cancellations presented in the original expression (6.10) for the Ursell coefficients. Inequality (6.24) yields bounds similar to Ruelle s ....

G. C. Rota. On the foundations of combinatorial theory I. Theory of Mobius functions. Wahrsch. Z. Verw. Geb., 2:340--368, 1964.

....tree graph in f1; 2; ng is a graph 2 GN such that j j = n Gamma 1 (roughly a graph with no loops) The number OE(f) appears naturally as the Ursell coefficent of the polymer gas Gibbs factor expf Gamma P i j U(R i ; R j )g. The formula (9) was originally proved by Rota in 1964 [10] (more recent proofs of the Rota inequality can be found in [5] 8] 11] and in [12] The Tree Graph Identity These combinatorial problems are due to the Mayer expansion mechanism. To be specific, the combinatorial difficulties arising when one tries to check the model dependent condition (8) ....

G. C Rota, On the foundations of the combinatorial theory. I Theory of Moebius functions, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 2, 340-368 (1964)

....an arrangement A, let (X) 0; X) denote the Mobius function of the lattice L(A) it is uniquely defined by X Y X (Y ) ffi 0;X where ffi 0;X is the Kronecker delta. The Mobius function is one of the fundamental invariants of any partially ordered set; see the seminal article of Rota [11]. The 1 characteristic polynomial of A is (A; t) X X2L(A) X)t dimX : 2) Since the characteristic polynomial is just the generating function for the Mobius function, it is also of prime importance. Our results in this paper give a combinatorial interpretation for the characteristic ....

....is ordered by reverse inclusion so that S Y X Y ae X. In particular g(R n ) j[ Gammas; s] n n S Aj. Note also that X [ Gammas; s] n is combinatorially just a cube of dimension dimX and side t so that f(X) t dimX . Finally, f(X) P Y X g(Y ) so by the Mobius Inversion Theorem [11] j[ Gammas; s] n n [ Aj = g( 0) X X2L(A) X)f(X) X X2L(A) X)t dimX = A; t) 3 which is the desired result. In the proof of Theorem 2.1, it was crucial that each of the subspaces X under consideration had exactly t dim(X) points in [ Gammas; s] n . In fact, the only ....

G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340--368.

....is also a multiplicative function. The sequence of numbers determining is found to be the one of the signed Catalan numbers, 0 n ; 1 n ] Gamma1) n Gamma1 (2n Gamma 2) n Gamma 1) n ; n 1 (2. 16) see [7] Section 7) For the general theory of the Moebius function on posets see [10], or [15] Chapter 3. As it was realized in [13] the convolution with plays an important role in the combinatorial approach to the theory of free random variables; this will be confirmed by the development presented in the next section (see the discussion in 3.2, 3.3) The formula which we ....

G.-C. Rota. On the foundations of combinatorial theory (I): Theory of Moebius functions, Z. Wahrscheinlichkeitstheorie verw. Geb. 2(1964), 340-368.

....student in a class, For a ring to pass the exam, it has to get 100 . Combinatorics has never fitted this pattern very well. When Gian Carlo Rota and various co workers wrote an influential series of papers with the title On the foundations of combinatorial theory in the 1960s and 1970s (see [27, 8], for example) one reviewer compared combinatorialists to nomads on the steppes who had not managed to construct the cities in which other mathematicians dwell, and expressed the hope that these papers would at least found a thriving settlement. While Rota s papers have been very influential, ....

G.-C. Rota, On the foundations of combinatorial theory, I: Theory of M obius functions, Zeitschrift f ur Wahrscheinlichkeitstheorie 2 (1964), 340--368. 9

.... 1 and an arti cial bottom element 0, the poset V (P ) becomes a lattice V (P ) Note that we adjoin 1 also in the case where V (P ) already has a top element corresponding to a face containing all vertices of P . For every element S 2 V (P ) we de ne the M obius function, see Rota [11] and Stanley [12] S) 8 : 1 if S = 0 ; X S 0 (S (S 0 ) otherwise : The M obius number (V (P ) 1) of V (P ) can be computed in time bounded polynomially in jV (P ) j. Since it is well known (see Stanley [12, 3.8.6] that (V (P ) e (V (P ) 1) this proves the ....

G.-C. Rota, On the foundations of combinatorial theory { I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340-368.

....only. Then F (L 1 ; LN L) f( L) 3. Let f (X; L) f (L 1 : LN X;L) the number of alignments of length L with at least the columns j 2 X consisting of blanks only. Then f (X; L) N Y i=1 L jXj L and f (X; L) X X Y f1; Lg f(Y; L) 4. By M obius inversion [5], this implies F (L 1 LN L) X x 0 ( 1) x L x N Y i=1 L x L ( The standard proof for this fact is the following: X x 0 ( 1) x L x N Y i=1 L x L i = X X f1; Lg ( 1) jXj f (X; L) X X f1; Lg ( 1) jXj X X Y f1; Lg f(Y; L) X Y ....

G.-C. Rota. On the foundations of combinatorial theory I. theory of M obius functions. Z. Wahrscheinlichkeitstheorie, 2:340--368, 1964.

....the integers. We also give some applications of this second main theorem, including the Tamari lattices. 1 Bounded below sets In a fundamental paper [25] Whitney showed how broken circuits could be used to compute the coefficients of the chromatic polynomial of a graph. In another seminal paper [20], Rota refined and extended Whitney s theorem to give a characterization of the Mobius function of a geometric lattice. Then one of us [21] generalized Rota s result to a larger class of lattices. In this paper we will present a theorem for an arbitrary finite lattice that includes all the others ....

.... Theta be any total order on A(L) Then each circuit C gives rise to a broken circuit C 0 = C n c where c is the first element of C under Theta. A set B A(L) is NBC (no broken circuit) if B does not contain any broken circuit and in this case B is an NBC base for x = W B. Rota s NBC theorem [20] is as follows. Theorem 1.2 (Rota) Let L be a finite geometric lattice and let Theta be any total order on A(L) Then for all x 2 L we have = Gamma1) ae(x) Delta (number of NBC bases of x) To derive this result from Theorem 1.1, we first prove that when L is geometric and Theta is ....

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G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340--368.

....from Gamma Omega n Delta to Q. Set A = L n0 V n , with multiplication defined as follows: for f 2 V n , g 2 Vm , and X 2 Gamma Omega n m Delta , fg) X) X Y 2( X n ) f(Y )g(X n Y ) This is the reduced incidence algebra of the poset of finite subsets of Omega (Rota [13]) It is a commutative and associative algebra with identity, but is far from an integral domain: any function with finite support is nilpotent. Now, if G is any permutation group on Omega Gamma let A G = L n0 V G n , where V G n consists of the functions in V n which are G invariant ....

G.-C. Rota, On the foundations of combinatorial theory, I: Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340--368.

....Theorem 2.3. If f : Z0 C then DeltaSf (n) f(n) One of the advantages of the combinatorial Mobius function is that its inversion theorem unifies and generalizes the previous three results. In addition, it makes the definition (2) transparent, encodes topological information about posets [6, 40], and has even been used to bound the running time of certain algorithms [9] We will now define this powerful invariant. Let finite P be a poset with partial order . If P has a unique minimal element then it will be denoted 0 = 0 P , and if it has a unique maximal element then we will use the ....

.... called because in I(D n ) it is related to the Riemann zeta function) given by i(x; y) 1 for all intervals [x; y] It is easy to see that i is invertible in I(P ) and in fact that i Gamma1 = where is defined by (4) The fundamental result about is the combinatorial Mobius Inversion Theorem [40]. Theorem 2.4. Let P be a finite poset and f; g : P C . 1. If for all x 2 P we have f(x) P yx g(y) then g(x) X yx (y; x)f(y) 2. If for all x 2 P we have f(x) P yx g(y) then g(x) X yx (x; y)f(y) It is now easy to obtain the Theorems 2.1, 2.2, and 2.3 as corollaries by ....

[Article contains additional citation context not shown here]

G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340--368.

....in a special case of Rota s Crosscut Theorem as well as in related proofs of Walker on Hall s Theorem and Reiner on characteristic and Poincar e polynomials. 1 Rota s theorem and its generalization One of the most beautiful and useful theorems in algebraic combinatorics is Rota s theorem [14] characterizing the Mobius function of a geometric lattice in terms of subsets of atoms which are NBC, i.e. contain no broken circuit. In this note we will generalize Rota s theorem to any lattice satisfying a simple condition and give applications to the weak Bruhat order of a Coxeter group and ....

....so is any B A(L) Furthermore, there are no circuits so any such B is NBC. Finally, independence of A(L) implies that B 6= B 0 for any B 6= B 0 . The corollary now follows from Theorem 1.2. 5 We note that Corollary 2. 1 also follows easily from a special case of Rota s Crosscut Theorem [14], proved by involutions in Section 3. We now derive the Mobius function of the weak Bruhat order of a Coxeter group which is a result of Bjorner [2] We do not consider the strong ordering because it is not a lattice in general. Any terminology from the theory of Coxeter groups not defined here ....

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G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340--368.

....; Lg of cardinality x L Gamma max(L 1 ; LN ) there exist f (X; L) f (L 1 ; LN ; X;L) N Y i=1 L Gamma x L i such alignments with at least all those columns consisting of blanks only which are indexed by elements j 2 X. Consequently, by Mobius inversion [6], the sum X 0xL Gammamax(L1 ; L N ) Gamma1) x L x N Y i=1 L Gamma x L i coincides with the number F (L 1 ; LN ; L) of all standard alignments of total length L without any column consisting of blanks only. Remark: The standard proof for this fact runs as follows: ....

G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 340-368 (1964).

....x x 0 . Thus, Rota s result follows from Theorem 2.1 # . 2 As a second application, we deduce a recent result of Blass and Sagan [1] on the Mobius function of a finite lattice. As pointed out in [1] it generalizes a particular case of Rota s broken circuit theorem on geometric lattices [10] as well as a prior generalization of that particular case due to Sagan [11] Corollary 2.5 (Blass Sagan) Let L be a finite lattice, whose set of atoms A(L) is given a partial order, which is denoted by E to distinguish it from the partial order # in L.LetX consist of all non empty subsets X of ....

G.C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie Verw. Geb. 2 (1964), 340--368.

....x x 0 . Thus, Rota s result follows from Theorem 2.1 0 . 2 As a second application, we deduce a recent result of Blass and Sagan [1] on the Mobius function of a finite lattice. As pointed out in [1] it generalizes a particular case of Rota s broken circuit theorem on geometric lattices [10] as well as a prior generalization of that particular case due to Sagan [11] Corollary 2.5 (Blass Sagan) Let L be a finite lattice, whose set of atoms A(L) is given a partial order, which is denoted by E to distinguish it from the partial order in L. Let X consist of all non empty subsets X ....

G.C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie Verw. Geb. 2 (1964), 340--368.

.... Phi Phi Phi Phi Phi Phi Phi Phi Phi r r r r r r r r a ba ab e aba bab b w o 1 2 4 1 2 3 4 4 1 4 1 Figure 5. Tilted Bruhat order D a (W ) for W of type B 2 Recall [21] that a finite graded poset with 0 and 1 (resp. with 0) is called Eulerian (resp. lower Eulerian) if its Mobius function [20] is given by (x; y) Gamma1) rank(y) Gammarank(x) for any x y. A well known (but non trivial cf. 2, 8, 18] theorem of Verma [23, 24] asserts that any interval in the Bruhat order of any Coxeter group is Eulerian. To our knowledge, no simple proof of this result is known, except for the ....

G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, in: Gian-Carlo Rota on combinatorics, Birkhauser, 1995, pp. 3-31.

....the enumeration of permutations with restricted position, and the enumeration of regions in hyperplane arrangements. 1. Introduction The Mobius function of a finite partially ordered set has been a pervasive theme in combinatorics ever since Rota s revolutionary paper Foundations I [21]. Its ubiquity is quite astonishing; it is related to such diverse topics as the four color theorem, the homology of simplicial complexes, and symmetric functions. Unexpected new applications are still being found today, e.g. Athanasiadis s finite field method for subspace arrangements [1] or ....

G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 2 (1964), 340--368.

.... above gets considerably worse when the line has a slope different from one (formula (6) and Theorem 2) Yet the proof of Theorem 2 shows that there is still enough symmetry left in the recurrence relation to allow significant simplifications. We enlist the help of the Finite Operator Calculus [3] to solve these problems. Suppose, B is a degree reducing linear operator on the algebra of polynomials, like the derivative operator D or the (forward) difference operator Delta. We are interested in polynomial sequences fp n (x)g n0 ; deg(p n ) n, which solve the system of operator ....

.... n l x l l (3) is the total weight of all such paths from the origin to (x; n) Note that this solution is symmetric in n and x for nonnegative integers, as required by the recursion (2) The theory of delta operators and Sheffer sequences was developed in the Finite Operator Calculus [3]; an overview of applications to the initial value problem Bp n (x) p n Gamma1 (x) for all n = 1; 2; 4) p n (cn ff) y n for all n = 0; 1; in combinatorics can be found in [2] In this paper we solve a recursive initial value problem (Theorem 1) where instead of given initial ....

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G.-C. Rota, D. Kahaner, and A. Odlyzko, On the foundations of combinatorial theory, VIII. Finite operator calculus, J. Math. Anal. Appl. 42 (1973) 684-760.

....map : H Phi Gamma S Phi ; x 7 x Phi is isotone and satisfies ffi = id as well as Phi 1 ffi ( Phi 1 ) Phi 1 . Hence ffi is a closure operator in the sense of combinatorics and j : ffi (S a ) H a is an isomorphism of posets. Now the lemma of Rota ([22]) tells us: X s 0 2Sa ; s 0 ) x Sa (s; s 0 ) 8 : Ha ( s) x) if s = ffi (s) 0 otherwise : 2) Lemma 3.1 If the characteristic p of F q does neither divide any jh Phi 1 i=h Phi 2 ij for Phi 2 Phi 1 2 S nor jT or(X=h Phii)j, then dim Fq hdff j ff 2 Phi 1 i = rankh Phi ....

....) Phi 1 ) Phi 2 ) w . If S Phi Bn ;w denotes the dual of S Phi Bn ;w (i.e. with reversed inclusion order) then the map d : S Phi Bn ;w Gamma S Phi Dn ;w ; Phi 1 7 Phi 1 Phi Dn together with the embedding Phi Dn , Phi Bn is a Galois connection (in the sense of [22]) Proof: It is clear that the assignment 7 Phi 2; is a right inverse of j . Moreover Phi 0 Phi 2; Phi 0 ) for all Phi 0 2 [ Phi 1 ; Phi 2 ] w . The rest is clear. ffi If Phi is indecomposable of type A then is an isomorphism of posets. Using 4.2 the determination of Mobius ....

G.C. Rota, On the foundations of combinatorial theory I. Mobius functions Z.f. Wahrscheinlichkeitstheorie 2 (1964), 340- 368

....upper bound of Proposition 3. We get the lower bound of Proposition 4 when we reduce the sequence equality function (see Section 3, Corollary 3) to undirected graph connectivity via a polynomial projection reduction. We can only give a very brief treatment on Mobius functions. For more see, e.g. [13]. Let S be a finite partially ordered set. The incidence algebra A(S) is defined as follows: Consider the set of functions of two variables f(x; y) for x and y ranging in S, having values in IR, the field of real numbers, and with the property that f(x; y) 0 whenever x 6 y. The sum and the ....

G.-C. Rota, On the foundation of combinatorial theory: I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2(1964), pp. 340--368.

....of Lemma 1. The first is used in the proof of the Theorem. The second is used for our results on packings in Section 5. In [4] we have developed a more general machinery to study partitions of products of hypergraphs. We employ there Rota s theorem of M obius transforms for posets (see [2] or [3] However, the best concrete results there are covered also by [1] and the present result. 3. Two consequences of Lemma 1 Lemma 2. Let H n be a product of d uniform hypergraphs with loops, then for a partition P of H n d n jPj = X I [n] v I 2V I (d Gamma 1) n GammajIj J ....

G.C. Rota, On the foundations of combinatorial theory I. M obius functions, Z. Wahrscheinlichkeitstheorie 2, 340--368, 1964.

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G.C. Rota, D. Kahaner and A. Odlyzko, On the foundations of combinatorial theory, VIII. Finite operator calculus, J. Math. Anal. Appl. 42, 1973, pp. 685--760.

.... in dealing with a sequence of numbers (such as the Bell numbers; see Rota [10] and for studying combinatorial algebraic objects like binomial sequences and algebraic invariants of polynomial systems (Hilbert [5] Kung and Rota [7] We refer to Roman and Rota [9] and Rota, Kahaner, and Odlyzko [11] for the history and development of Umbral Calculus. An algebraic treatment was given in Joni and Rota [6] New developments can be found in Rota and Taylor [13] and Rota, Shen, and Taylor [12] Also see Shen [14] for a recent application in wavelet analysis. In this paper, we shall employ the ....

G.-C. Rota, D. Kahaner, and A. Odlyzko, On the foundations of combinatorial theory. VIII. Finite operator calculus, J. Math. Anal. Appl. 42 (1973), 684#760.

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G. C. Rota, "On the foundations of combinatorial theory," Zeitschrift Fur Wahrscheinlichkeitstheorie Und Verwandte Gebietee, vol. 2, pp. 340--368, 1964.

No context found.

G. C. Rota, "On the foundations of combinatorial theory," Zeitschrift Fur Wahrscheinlichkeitstheorie Und Verwandte Gebietee, vol. 2, pp. 340--368, 1964.

No context found.

G. C. Rota, "On the foundations of combinatorial theory," Zeitschrift Fur Wahrscheinlichkeitstheorie Und Verwandte Gebietee, vol. 2, pp. 340--368, 1964.

No context found.

G.-C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340--368.

No context found.

G.-C. Rota, D. Kahaner, and A. Odlyzko, On the foundations of combinatorial theory. VIII. Finite operator calculus, J. Math. Anal. Appl. 42 (1973) 684--760. Royal Institute of Technology, S-100 44 Stockholm, Sweden, jrge@math.kth.se 5

No context found.

G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340--368.

No context found.

G.-C. Rota, On the foundations of combinatorial theory, I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie u. verw. Geb. 2 (1964), 340 -- 368.

No context found.

G.-C. Rota, D. Kahaner, and A. Odlyzko, On the foundations of combinatorial theory. VIII. Finite operator calculus, J. Math. Anal. Appl. 42 (1973) 684--760. Royal Institute of Technology, S-100 44 Stockholm, Sweden, jrge@math.kth.se 5

No context found.

Rota, G.-C., Kahaner, D. and Odlyzko, A. (1973). On the foundations of combinatorial theory, VIII. Finite operator calculus, J. Math. Anal. Appl. 42, 684-760.

No context found.

G.-C. Rota, On the foundations of combinatorial theory, I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie u. verw. Geb. 2 (1964), 340--368.

No context found.

G.C. Rota. On the foundations of combinatorial theory I. Theory of Mobius functions. Z. Wahrscheinlichkeitsrechnung, 2 (1964), 340--368.

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G.-C. Rota, On the foundations of combinatorial theory: I. Theory of Mobius functions. Z. Wahrsch. verw. Gebiete 2 (1964), 340-368. MR 30 #4688.

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