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{"Title": "Carlitz-Riordan $q$-Catalan numbers $C_n(q)$", "Definition": "The Carlitz-Riordan $q$-Catalan polynomial $C_n(q)$ CITE{CarlitzRiordan} is $C_n(q,t)$ at $t=1$. Equivalently, it is the area generating function of Dyck paths of semilength $n$ in the convention of CITE{statistics}.", "Keywords": ["Carlitz-Riordan q-Catalan numbers", "Carlitz Riordan q-Catalan numbers", "q-Catalan numbers", "Dyck paths", "area statistic", "dinv statistic"], "Tags": ["polynomial", "combinatorics"], "Parameters": {"n": {"type": "Z", "title": "semilength", "display": "$n$", "constraints": "a nonnegative integer"}}, "Display properties": {"number-header": "$C_n(q)$"}, "Comments": {"statistics": "For $n=0$ the area sequence is empty. For $n>0$, an area sequence is $(a_1,\\ldots,a_n)$ with $a_1=0$ and $0\\leq a_{i+1}\\leq a_i+1$ for $1\\leq i<n$. Its area is $\\sum_i a_i$, and its dinv is the number of pairs $i<j$ with $a_i-a_j\\in\\{0,1\\}$."}, "Formulas": {"formula-carlitz": "$C_n(q)=C_n(q,1)=\\sum_D q^{\\operatorname{dinv}(D)}=\\sum_D q^{\\operatorname{area}(D)}$, summed over Dyck paths $D$ of semilength $n$ CITE{Haglund}.", "formula-catalan": "$C_n(1)=\\frac{1}{n+1}\\binom{2n}{n}$.", "formula-small": "$C_0(q)=C_1(q)=1$ and $C_2(q)=q+1$."}, "Similar tables": [{"table": "HREF{T262}[$q,t$-Catalan numbers]", "relation": "stores the two-variable polynomial whose $t=1$ specialisation is this table. This table is the likely target for a reader holding the one-variable area or dinv distribution; a reader holding the full dinv-area polynomial wants T262."}, {"table": "HREF{T265}[MacMahon $q$-Catalan numbers]", "relation": "stores the other standard one-variable $q$-Catalan polynomial, $q^{\\binom n2}C_n(q,q^{-1})$, while this table stores $C_n(q,1)$. Neither one-variable table determines the full $C_n(q,t)$."}], "Links": {"SageCatalan": {"title": "SageMath: q-analogues", "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/q_analogues.html#sage.combinat.q_analogues.qt_catalan_number"}}, "References": {"Haglund": {"bib": "J. Haglund, The q,t-Catalan Numbers and the Space of Diagonal Harmonics, University Lecture Series 41, American Mathematical Society, Providence, RI, 2008."}, "GarsiaHaiman": {"bib": "A. M. Garsia and M. Haiman, A remarkable q,t-Catalan sequence and q-Lagrange inversion, J. Algebraic Combin. 5 (1996), 191-244."}, "CarlitzRiordan": {"bib": "L. Carlitz and J. Riordan, Two element lattice permutation numbers and their q-generalization, Duke Math. J. 31 (1964), 371-388."}, "MacMahon": {"bib": "P. A. MacMahon, Combinatory Analysis, Vol. II, Cambridge University Press, Cambridge, 1916."}}, "Programs": {"program-sage": {"language": "Sage", "code": "from sage.combinat.q_analogues import qt_catalan_number\n\nqt_catalan_number(12).subs(t=1)"}}, "Data properties": {"type": "Z[]", "rigour": "exact", "complete": "no", "complete-note": "it holds the rows for $3\\leq n\\leq11$; the omitted rows with $n\\leq2$ are the closed forms in CITE{formula-small}", "rigour details": "Every entry is an exact polynomial with integer coefficients. The generator computes $C_n(q,t)$ with Sage's `qt_catalan_number` CITE{SageCatalan} and then specialises at $t=1$.\n\nBefore the values were written, the rows were checked against an independent enumeration of Dyck-path area sequences using the dinv and area statistics. The checks also verified CITE{formula-catalan} and the small cases in CITE{formula-small} over the stored range."}, "Numbers": {"3": "q^3 + q^2 + 2*q + 1", "4": "q^6 + q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 3*q + 1", "5": "q^10 + q^9 + 2*q^8 + 3*q^7 + 5*q^6 + 5*q^5 + 7*q^4 + 7*q^3 + 6*q^2 + 4*q + 1", "6": "q^15 + q^14 + 2*q^13 + 3*q^12 + 5*q^11 + 7*q^10 + 9*q^9 + 11*q^8 + 14*q^7 + 16*q^6 + 16*q^5 + 17*q^4 + 14*q^3 + 10*q^2 + 5*q + 1", "7": "q^21 + q^20 + 2*q^19 + 3*q^18 + 5*q^17 + 7*q^16 + 11*q^15 + 13*q^14 + 18*q^13 + 22*q^12 + 28*q^11 + 32*q^10 + 37*q^9 + 40*q^8 + 44*q^7 + 43*q^6 + 40*q^5 + 35*q^4 + 25*q^3 + 15*q^2 + 6*q + 1", "8": "q^28 + q^27 + 2*q^26 + 3*q^25 + 5*q^24 + 7*q^23 + 11*q^22 + 15*q^21 + 20*q^20 + 26*q^19 + 34*q^18 + 42*q^17 + 53*q^16 + 63*q^15 + 73*q^14 + 85*q^13 + 96*q^12 + 106*q^11 + 113*q^10 + 118*q^9 + 118*q^8 + 115*q^7 + 102*q^6 + 86*q^5 + 65*q^4 + 41*q^3 + 21*q^2 + 7*q + 1", "9": "q^36 + q^35 + 2*q^34 + 3*q^33 + 5*q^32 + 7*q^31 + 11*q^30 + 15*q^29 + 22*q^28 + 28*q^27 + 38*q^26 + 48*q^25 + 63*q^24 + 77*q^23 + 97*q^22 + 116*q^21 + 139*q^20 + 162*q^19 + 190*q^18 + 215*q^17 + 245*q^16 + 268*q^15 + 293*q^14 + 314*q^13 + 331*q^12 + 338*q^11 + 338*q^10 + 326*q^9 + 303*q^8 + 268*q^7 + 219*q^6 + 167*q^5 + 112*q^4 + 63*q^3 + 28*q^2 + 8*q + 1", "10": "q^45 + q^44 + 2*q^43 + 3*q^42 + 5*q^41 + 7*q^40 + 11*q^39 + 15*q^38 + 22*q^37 + 30*q^36 + 40*q^35 + 52*q^34 + 69*q^33 + 87*q^32 + 111*q^31 + 138*q^30 + 171*q^29 + 207*q^28 + 249*q^27 + 295*q^26 + 348*q^25 + 405*q^24 + 466*q^23 + 531*q^22 + 598*q^21 + 665*q^20 + 734*q^19 + 801*q^18 + 862*q^17 + 918*q^16 + 958*q^15 + 990*q^14 + 1003*q^13 + 995*q^12 + 959*q^11 + 901*q^10 + 813*q^9 + 704*q^8 + 574*q^7 + 434*q^6 + 301*q^5 + 182*q^4 + 92*q^3 + 36*q^2 + 9*q + 1", "11": "q^55 + q^54 + 2*q^53 + 3*q^52 + 5*q^51 + 7*q^50 + 11*q^49 + 15*q^48 + 22*q^47 + 30*q^46 + 42*q^45 + 54*q^44 + 73*q^43 + 93*q^42 + 121*q^41 + 152*q^40 + 193*q^39 + 237*q^38 + 295*q^37 + 356*q^36 + 431*q^35 + 513*q^34 + 611*q^33 + 714*q^32 + 837*q^31 + 964*q^30 + 1109*q^29 + 1257*q^28 + 1422*q^27 + 1588*q^26 + 1770*q^25 + 1947*q^24 + 2131*q^23 + 2307*q^22 + 2481*q^21 + 2636*q^20 + 2784*q^19 + 2900*q^18 + 2990*q^17 + 3037*q^16 + 3039*q^15 + 2992*q^14 + 2887*q^13 + 2717*q^12 + 2486*q^11 + 2203*q^10 + 1871*q^9 + 1515*q^8 + 1149*q^7 + 806*q^6 + 512*q^5 + 282*q^4 + 129*q^3 + 45*q^2 + 10*q + 1"}}