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Natural endomorphisms of quasishuffle Hopf algebras
 ha l0 0, v er sio n  6 N ov DEFORMATIONS OF SHUFFLES AND QUASISHUFFLES 23
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THE ALGEBRAIC COMBINATORICS OF SNAKES
, 2012
"... Snakes are analogues of alternating permutations defined for any Coxeter group. We study these objects from the point of view of combinatorial Hopf algebras, such as noncommutative symmetric functions and their generalizations. The main purpose is to show that several properties of the generating ..."
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Snakes are analogues of alternating permutations defined for any Coxeter group. We study these objects from the point of view of combinatorial Hopf algebras, such as noncommutative symmetric functions and their generalizations. The main purpose is to show that several properties of the generating functions of snakes, such as differential equations or closed form as trigonometric functions, can be lifted at the level of noncommutative symmetric functions or free quasisymmetric functions. The results take the form of algebraic identities for type B noncommutative symmetric functions, noncommutative supersymmetric functions and colored free quasisymmetric functions.
Superization and (q, t)specialization in combinatorial Hopf algebras
, 2009
"... We extend a classical construction on symmetric functions, the superization process, to several combinatorial Hopf algebras, and obtain analogs of the hookcontent formula for the (q,t)specializations of various bases. Exploiting the dendriform structures yields in particular (q,t)analogs of the ..."
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We extend a classical construction on symmetric functions, the superization process, to several combinatorial Hopf algebras, and obtain analogs of the hookcontent formula for the (q,t)specializations of various bases. Exploiting the dendriform structures yields in particular (q,t)analogs of the BjörnerWachs qhooklength formulas for
Unital versions of the higher order peak algebras
"... Abstract. We construct unital extensions of the higher order peak algebras defined by Krob and the third author in [Ann. Comb. 9 (2005), 411–430.], and show that they can be obtained as homomorphic images of certain subalgebras of the MantaciReutenauer algebras of type B. This generalizes a result ..."
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Abstract. We construct unital extensions of the higher order peak algebras defined by Krob and the third author in [Ann. Comb. 9 (2005), 411–430.], and show that they can be obtained as homomorphic images of certain subalgebras of the MantaciReutenauer algebras of type B. This generalizes a result of Bergeron, Nyman and the first author [Trans. AMS 356 (2004), 2781–2824.]. 1.
Enumeration and Random Generation of Concurrent Computations †
"... In this paper, we study the shuffle operator on concurrent processes (represented as trees) using analytic combinatorics tools. As a first result, we show that the mean width of shuffle trees is exponentially smaller than the worst case upperbound. We also study the expected size (in total number of ..."
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In this paper, we study the shuffle operator on concurrent processes (represented as trees) using analytic combinatorics tools. As a first result, we show that the mean width of shuffle trees is exponentially smaller than the worst case upperbound. We also study the expected size (in total number of nodes) of shuffle trees. We notice, rather unexpectedly, that only a small ratio of all nodes do not belong to the last two levels. We also provide a precise characterization of what “exponential growth ” means in the case of the shuffle on trees. Two practical outcomes of our quantitative study are presented: (1) a lineartime algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random generation of concurrent runs.
THE (1 − E)TRANSFORM IN COMBINATORIAL HOPF ALGEBRAS
, 2009
"... We extend to several combinatorial Hopf algebras the endomorphism of symmetric functions sending the first powersum to zero and leaving the other ones invariant. As a “transformation of alphabets”, this is the (1 − E)transform, where E is the “exponential alphabet”, whose elementary symmetric fun ..."
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We extend to several combinatorial Hopf algebras the endomorphism of symmetric functions sending the first powersum to zero and leaving the other ones invariant. As a “transformation of alphabets”, this is the (1 − E)transform, where E is the “exponential alphabet”, whose elementary symmetric functions are. In the case of noncommutative symmetric functions, we recover Schocker’s idempotents for derangement numbers [Discr. Math. 269 (2003), 239]. From these idempotents, we construct subalgebras of the descent algebras analogous to the peak algebras and study their representation theory. The case of WQSym leads to similar subalgebras of the SolomonTits algebras. In FQSym, the study of the transformation boils down to a simple solution of the Tsetlin library in the uniform case.
Representation theory of the higher order peak algebras, preprint math.CO/0906.5236
"... Abstract. The representation theory (idempotents, quivers, Cartan invariants and Loewy series) of the higher order unital peak algebras is investigated. On the way, we obtain new interpretations and generating functions for the idempotents of descent algebras introduced in [F. Saliola, J. Algebra 32 ..."
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Abstract. The representation theory (idempotents, quivers, Cartan invariants and Loewy series) of the higher order unital peak algebras is investigated. On the way, we obtain new interpretations and generating functions for the idempotents of descent algebras introduced in [F. Saliola, J. Algebra 320 (2008) 3866.] 1.
INVERSION OF SOME SERIES OF FREE QUASISYMMETRIC FUNCTIONS
"... Abstract. We give a combinatorial formula for the inverses of the alternating sums of free quasisymmetric functions of the form F ω(I) where I runs over compositions with parts in a prescribed set C. This proves in particular three special cases (no restriction, even parts, and all parts equal to 2 ..."
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Abstract. We give a combinatorial formula for the inverses of the alternating sums of free quasisymmetric functions of the form F ω(I) where I runs over compositions with parts in a prescribed set C. This proves in particular three special cases (no restriction, even parts, and all parts equal to 2) which were conjectured by B. C. V. Ung in [Proc. FPSAC’98, Toronto]. hal00325271, version 1 26 Sep 2008