The doubloon polynomial triangle
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The doubloon polynomial triangle Dominique Foata and Guo-Niu Han Dedicated to George Andrews, on the occasion of his seventieth birthday. ABSTRACT. The doubloon polynomials are generating functions for a class of combinatorial objects called normalized doubloons by the com- pressed major index. They provide a refinement of the q-tangent numbers and also involve two major specializations: the Poupard triangle and the Catalan triangle. 1. Introduction The doubloon (?) polynomials dn,j(q) (n ≥ 1, 2 ≤ j ≤ 2n) introduced in this paper serve to globalize the Poupard triangle [Po89] and the classical Catalan triangle [Sl07]. They also provide a refinement of the q-tangent numbers, fully studied in our previous paper [FH08]. Finally, as generating polynomials for the doubloon model, they constitute a common combina- torial set-up for the above integer triangles. They may be defined by the following recurrence: (D1) d0,j(q) = ?1,j (Kronecker symbol); (D2) dn,j(q) = 0 for n ≥ 1 and j ≤ 1 or j ≥ 2n + 1; (D3) dn,2(q) = ∑ j qj?1 dn?1,j(q) for n ≥ 1; (D4) dn,j(q)? 2 dn,j?1(q) + dn,j?2(q) = ?(1? q) j?3 ∑ i=1 qn+i+1?j
Voir
- doubloon polynomials
- called normalized
- following recurrence
- previ- ous paper
- normalized doubloon
- excellent commented bibliography
- catalan triangle
- major specializations
The doubloon polynomial triangle
Dominique Foata and Guo-Niu Han
Dedicated to George Andrews, on the occasion of his seventieth birthday.
ATCARTSB doubloon polynomials are generating functions for. The a class of combinatorial objects called normalized doubloons by the com-pressed major index. They provide a refinement of theq-tangent numbers and also involve two major specializations: the Poupard triangle and the Catalan triangle.
1. Introduction Thedoubloon(∗)polynomialsdnj(q) (n≥12≤j≤2n) introduced in this paper serve to globalize thePoupard triangle[Po89] and the classical Catalan triangle[Sl07]. They also provide a refinement of theq-tangent numbers, fully studied in our previous paper [FH08]. as generating Finally, polynomials for the doubloon model, they constitute a common combina-torial set-up for the above integer triangles. They may be defined by the following recurrence: (D1)d0j(q) =δ1j(Kronecker symbol); (D2)dnj(q) = 0 forn≥1 andj≤1 orj≥2n+ 1; (D3)dn2(q) =Pqj−1dn−1j(q) forn≥1; j (D4)dnj(q)−2dnj−1(q) +dnj−2(q) j−3 =−(1−q)Pqn+i+1−jdn−1i(q) i=1 2n−1 −(1 +qn−1)dn−1j−2(q) + (1−q)Pqi−j+1dn−1i(q) i=j−1 forn≥2 and 3≤j≤2n. The polynomialsdnj(q) (n≥12≤j≤2n) are easily evaluated using (D1)–(D4) and form thedoubloon polynomial triangle, as shown in Fig. 1.1 . and 11′d12(q) d22(q)d23(q)d24(q) d3 2(q)d33(q)d34(q)d35(q)d36(q) d42(q)d43(q)d44(q)d45(q)d46(q)d47(q)d48(q) Fig. 1.1. The doubloon polynomial triangle
Key words and phrases.Doubloon polynomials, doubloon polynomial triangle, Poupard triangle, Catalan triangle, reduced tangent numbers. Mathematics Subject Classifications.05A15, 05A30, 33B10 (∗originally refers to a Spanish gold coin, the word “doubloon” ) Although it is here used to designate a permutation written as a two-row matrix.
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d12(q) = 1;d22(q) =q;d23(q) =q+ 1;d24(q) = 1; d32(q) = 2q3+2q2;d33(q) = 2q3+4q2+2q;d34(q) =q3+4q2+4q+1; d35(q) = 2q2+ 4q+ 2;d36(q) = 2q+ 2; d42(q) = 5q6+12q5+12q4+5q3;d43(q) = 5q6+17q5+24q4+17q3+5q2; d44(q) = 3q6+ 15q5+ 29q4+ 29q3+ 15q2+ 3q; d45(q) =q6+ 9q5+ 25q4+ 34q3+ 25q2+ 9q+ 1; d46(q) = 3q5+ 15q4+ 29q3+ 29q2+ 15q+ 3; d47(q) = 5q4+ 17q3+ 24q2+ 17q+ 5;d48(q) = 5q3+ 12q2+ 12q+ 5. Fig. 11′ first doubloon polynomials. The Notice the different symmetries of the coefficients of the polynomials dnj(q), which will be fully exploited in Section 4 (Corollaries 4.3, 4.7, 4.8). Various specializations are displayed in Fig. 1.2 below, whereCn= 2n+12nstands for the celebrated Catalan number andtnfor thereduced n tangent numberoccurring in the Taylor expansion 2 tan(u2) =X(2un2n++11)!tn n≥0 (11) = 1u!1+u3!31 +u554+!u77+!43u9!9496 +u11111!6501+ The symbol Σ attached to each vertical arrow has the meaning “make the summation overj” anddn(q) is the polynomial further defined in (1.3). dnj(0)q←=−0dnj(q)q−=→1dnj(1) yΣyΣyΣ Cqn←=−0dn(q)q−=→1tn Fig. 1.2. The specializations ofdnj(q) Whenq= 1, the (D1)–(D4) recurrence becomes (P1)d0j(1) =δ0j(Kronecker symbol); (P2)dnj(1) = 0 forn≥1 andj≤1 orj≥2n+ 1; (P3)dn2(1) =Pdn−1j(1) forn≥1; j (P4)dnj(1)−2dnj−1(1) +dnj−2(1) =−2dn−1j−2(1) forn≥2 and 3≤j≤2n, which is exactly the recurrence introduced by Christiane Poupard [Po89]. 1 1 2 1 4 8 10 8 4 34 68 94 104 94 68 34 Fig. 1.3. The Poupard triangle (dnj(1))
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