The doubloon polynomial triangle

Dominique Foata

Description

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


• called normalized


• following recurrence


• previ- ous paper


• normalized doubloon


• excellent commented bibliography


• catalan triangle


• major specializations

Sujets

• Poupard
• Andrews
• Deutsch
• Callan
• Encyclopédia
• Taylor

Informations

• Publié par: pefav
• Nombre de lectures: 10
• Langue: English

Extrait

The doubloon polynomial triangle

Dominique Foata and Guo-Niu Han

Dedicated to George Andrews, on the occasion of his seventieth birthday.

A TCARTSB  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( ∗ )polynomialsd n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)d 0j (q) =δ 1j (Kronecker symbol); (D2)d nj (q) = 0 forn≥1 andj≤1 orj≥2n+ 1; (D3)d n2 (q) = P q j−1 d n−1j (q) forn≥1; j (D4)d nj (q)−2d nj−1 (q) +d nj−2 (q) j−3 =−(1−q) P q n+i+1−j d n−1i (q) i=1 2n−1 −(1 +q n−1 )d n−1j−2 (q) + (1−q) P q i−j+1 d n−1i (q) i=j−1 forn≥2 and 3≤j≤2n. The polynomialsd n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 ′ d 12 (q) d 22 (q)d 23 (q)d 24 (q) d 3 2 (q)d 33 (q)d 34 (q)d 35 (q)d 36 (q)  d 42 (q)d 43 (q)d 44 (q)d 45 (q)d 46 (q)d 47 (q)d 4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.

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d 12 (q) = 1;d 22 (q) =q;d 23 (q) =q+ 1;d 24 (q) = 1; d 32 (q) = 2q 3 +2q 2 ;d 33 (q) = 2q 3 +4q 2 +2q;d 34 (q) =q 3 +4q 2 +4q+1; d 35 (q) = 2q 2 + 4q+ 2;d 36 (q) = 2q+ 2; d 42 (q) = 5q 6 +12q 5 +12q 4 +5q 3 ;d 43 (q) = 5q 6 +17q 5 +24q 4 +17q 3 +5q 2 ; d 44 (q) = 3q 6 + 15q 5 + 29q 4 + 29q 3 + 15q 2 + 3q; d 45 (q) =q 6 + 9q 5 + 25q 4 + 34q 3 + 25q 2 + 9q+ 1; d 46 (q) = 3q 5 + 15q 4 + 29q 3 + 29q 2 + 15q+ 3; d 47 (q) = 5q 4 + 17q 3 + 24q 2 + 17q+ 5;d 48 (q) = 5q 3 + 12q 2 + 12q+ 5. Fig. 11 ′  first doubloon polynomials. The Notice the different symmetries of the coefficients of the polynomials d n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, whereC n = 2n+1  2 n  stands for the celebrated Catalan number andt n for thereduced n tangent numberoccurring in the Taylor expansion 2 tan(u2) = X (2 u n 2n + + 1 1 )! t n n≥0 (11) =  1 u ! 1+ u 3! 3 1 + u 5 5 4+ ! u 7 7 + ! 43 u 9! 9 496 + u 1 1 1 1 1 ! 6501+   The symbol Σ attached to each vertical arrow has the meaning “make the summation overj” andd n (q) is the polynomial further defined in (1.3). d nj (0) q ← = − 0 d nj (q) q − = → 1 d nj (1)  y Σ  y Σ  y Σ C q n ← = − 0 d n (q) q − = → 1 t n Fig. 1.2. The specializations ofd nj (q) Whenq= 1, the (D1)–(D4) recurrence becomes (P1)d 0j (1) =δ 0j (Kronecker symbol); (P2)d nj (1) = 0 forn≥1 andj≤1 orj≥2n+ 1; (P3)d n2 (1) = P d n−1j (1) forn≥1; j (P4)d nj (1)−2d nj−1 (1) +d nj−2 (1) =−2d n−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 (d nj (1))

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