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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) (n12j2n) 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 forn1 andj1 orj2n+ 1; (D3)dn2(q) =Pqj1dn1j(q) forn1; j (D4)dnj(q)2dnj1(q) +dnj2(q) j3 =(1q)Pqn+i+1jdn1i(q) i=1 2n1 (1 +qn1)dn1j2(q) + (1q)Pqij+1dn1i(q) i=j1 forn2 and 3j2n. The polynomialsdnj(q) (n12j2n) are easily evaluated using (D1)–(D4) and form thedoubloon polynomial triangle, as shown in Fig. 1.1 . and 11d12(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 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+12nstands for the celebrated Catalan number andtnfor thereduced n tangent numberoccurring in the Taylor expansion 2 tan(u2) =X(2un2n++11)!tn n0 (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). dnj(0)q=0dnj(q)q=1dnj(1) yΣyΣyΣ 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 forn1 andj1 orj2n+ 1; (P3)dn2(1) =Pdn1j(1) forn1; j (P4)dnj(1)2dnj1(1) +dnj2(1) =2dn1j2(1) forn2 and 3j2n, 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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