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

doubloon polynomials

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normalized doubloon

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catalan 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.

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(∗)polynomialsdnj(q) (n≥12≤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)d0j(q) =δ1j(Kronecker symbol); (D2)dnj(q) = 0 forn≥1 andj≤1 orj≥2n+ 1; (D3)dn2(q) =Pqj−1dn−1j(q) forn≥1; j (D4)dnj(q)−2dnj−1(q) +dnj−2(q) j−3 =−(1−q)Pqn+i+1−jdn−1i(q) i=1 2n−1 −(1 +qn−1)dn−1j−2(q) + (1−q)Pqi−j+1dn−1i(q) i=j−1 forn≥2 and 3≤j≤2n. The polynomialsdnj(q) (n≥12≤j≤2n) are easily evaluated using (D1)–(D4) and form thedoubloon polynomial triangle, as shown in Fig. 1.1 . and 11′d12(q) d22(q)d23(q)d24(q) d3 2(q)d33(q)d34(q)d35(q)d36(q)  d42(q)d43(q)d44(q)d45(q)d46(q)d47(q)d48(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.

d12(q) = 1;d22(q) =q;d23(q) =q+ 1;d24(q) = 1; d32(q) = 2q3+2q2;d33(q) = 2q3+4q2+2q;d34(q) =q3+4q2+4q+1; d35(q) = 2q2+ 4q+ 2;d36(q) = 2q+ 2; d42(q) = 5q6+12q5+12q4+5q3;d43(q) = 5q6+17q5+24q4+17q3+5q2; d44(q) = 3q6+ 15q5+ 29q4+ 29q3+ 15q2+ 3q; d45(q) =q6+ 9q5+ 25q4+ 34q3+ 25q2+ 9q+ 1; d46(q) = 3q5+ 15q4+ 29q3+ 29q2+ 15q+ 3; d47(q) = 5q4+ 17q3+ 24q2+ 17q+ 5;d48(q) = 5q3+ 12q2+ 12q+ 5. Fig. 11′ ﬁrst doubloon polynomials. The Notice the diﬀerent symmetries of the coeﬃcients of the polynomials dnj(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+12nstands for the celebrated Catalan number andtnfor thereduced n tangent numberoccurring in the Taylor expansion 2 tan(u2) =X(2un2n++11)!tn n≥0 (11) = 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). dnj(0)q←=−0dnj(q)q−=→1dnj(1) yΣyΣyΣ Cqn←=−0dn(q)q−=→1tn Fig. 1.2. The specializations ofdnj(q) Whenq= 1, the (D1)–(D4) recurrence becomes (P1)d0j(1) =δ0j(Kronecker symbol); (P2)dnj(1) = 0 forn≥1 andj≤1 orj≥2n+ 1; (P3)dn2(1) =Pdn−1j(1) forn≥1; j (P4)dnj(1)−2dnj−1(1) +dnj−2(1) =−2dn−1j−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 (dnj(1))

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