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MacMahon $q$-Catalan numbers $\widetilde C_n(q)$
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Polynomials
$n$ 
$\widetilde C_n(q)$
3:
q^6 + q^4 + q^3 + q^2 + 1
4:
q^12 + q^10 + q^9 + 2*q^8 + q^7 + 2*q^6 + q^5 + 2*q^4 + q^3 + q^2 + 1
5:
q^20 + q^18 + q^17 + 2*q^16 + 2*q^15 + 3*q^14 + 2*q^13 + 4*q^12 + 3*q^11 + 4*q^10 + 3*q^9 + 4*q^8 + 2*q^7 + 3*q^6 + 2*q^5 + 2*q^4 + q^3 + q^2 + 1
6:
q^30 + q^28 + q^27 + 2*q^26 + 2*q^25 + 4*q^24 + 3*q^23 + 5*q^22 + 5*q^21 + 7*q^20 + 6*q^19 + 9*q^18 + 7*q^17 + 9*q^16 + 8*q^15 + 9*q^14 + 7*q^13 + 9*q^12 + 6*q^11 + 7*q^10 + 5*q^9 + 5*q^8 + 3*q^7 + 4*q^6 + 2*q^5 + 2*q^4 + q^3 + q^2 + 1
7:
q^42 + q^40 + q^39 + 2*q^38 + 2*q^37 + 4*q^36 + 4*q^35 + 6*q^34 + 6*q^33 + 9*q^32 + 9*q^31 + 13*q^30 + 12*q^29 + 16*q^28 + 16*q^27 + 19*q^26 + 18*q^25 + 22*q^24 + 20*q^23 + 23*q^22 + 21*q^21 + 23*q^20 + 20*q^19 + 22*q^18 + 18*q^17 + 19*q^16 + 16*q^15 + 16*q^14 + 12*q^13 + 13*q^12 + 9*q^11 + 9*q^10 + 6*q^9 + 6*q^8 + 4*q^7 + 4*q^6 + 2*q^5 + 2*q^4 + q^3 + q^2 + 1
8:
q^56 + q^54 + q^53 + 2*q^52 + 2*q^51 + 4*q^50 + 4*q^49 + 7*q^48 + 7*q^47 + 10*q^46 + 11*q^45 + 16*q^44 + 16*q^43 + 22*q^42 + 23*q^41 + 29*q^40 + 30*q^39 + 37*q^38 + 37*q^37 + 45*q^36 + 45*q^35 + 51*q^34 + 51*q^33 + 58*q^32 + 55*q^31 + 61*q^30 + 58*q^29 + 62*q^28 + 58*q^27 + 61*q^26 + 55*q^25 + 58*q^24 + 51*q^23 + 51*q^22 + 45*q^21 + 45*q^20 + 37*q^19 + 37*q^18 + 30*q^17 + 29*q^16 + 23*q^15 + 22*q^14 + 16*q^13 + 16*q^12 + 11*q^11 + 10*q^10 + 7*q^9 + 7*q^8 + 4*q^7 + 4*q^6 + 2*q^5 + 2*q^4 + q^3 + q^2 + 1
9:
q^72 + q^70 + q^69 + 2*q^68 + 2*q^67 + 4*q^66 + 4*q^65 + 7*q^64 + 8*q^63 + 11*q^62 + 12*q^61 + 18*q^60 + 19*q^59 + 26*q^58 + 29*q^57 + 37*q^56 + 40*q^55 + 51*q^54 + 54*q^53 + 66*q^52 + 71*q^51 + 83*q^50 + 88*q^49 + 103*q^48 + 106*q^47 + 120*q^46 + 125*q^45 + 138*q^44 + 140*q^43 + 154*q^42 + 153*q^41 + 165*q^40 + 164*q^39 + 172*q^38 + 168*q^37 + 176*q^36 + 168*q^35 + 172*q^34 + 164*q^33 + 165*q^32 + 153*q^31 + 154*q^30 + 140*q^29 + 138*q^28 + 125*q^27 + 120*q^26 + 106*q^25 + 103*q^24 + 88*q^23 + 83*q^22 + 71*q^21 + 66*q^20 + 54*q^19 + 51*q^18 + 40*q^17 + 37*q^16 + 29*q^15 + 26*q^14 + 19*q^13 + 18*q^12 + 12*q^11 + 11*q^10 + 8*q^9 + 7*q^8 + 4*q^7 + 4*q^6 + 2*q^5 + 2*q^4 + q^3 + q^2 + 1
10:
q^90 + q^88 + q^87 + 2*q^86 + 2*q^85 + 4*q^84 + 4*q^83 + 7*q^82 + 8*q^81 + 12*q^80 + 13*q^79 + 19*q^78 + 21*q^77 + 29*q^76 + 33*q^75 + 43*q^74 + 48*q^73 + 62*q^72 + 68*q^71 + 85*q^70 + 94*q^69 + 113*q^68 + 124*q^67 + 148*q^66 + 160*q^65 + 186*q^64 + 201*q^63 + 229*q^62 + 244*q^61 + 276*q^60 + 290*q^59 + 322*q^58 + 337*q^57 + 368*q^56 + 381*q^55 + 412*q^54 + 421*q^53 + 449*q^52 + 456*q^51 + 480*q^50 + 481*q^49 + 502*q^48 + 497*q^47 + 512*q^46 + 504*q^45 + 512*q^44 + 497*q^43 + 502*q^42 + 481*q^41 + 480*q^40 + 456*q^39 + 449*q^38 + 421*q^37 + 412*q^36 + 381*q^35 + 368*q^34 + 337*q^33 + 322*q^32 + 290*q^31 + 276*q^30 + 244*q^29 + 229*q^28 + 201*q^27 + 186*q^26 + 160*q^25 + 148*q^24 + 124*q^23 + 113*q^22 + 94*q^21 + 85*q^20 + 68*q^19 + 62*q^18 + 48*q^17 + 43*q^16 + 33*q^15 + 29*q^14 + 21*q^13 + 19*q^12 + 13*q^11 + 12*q^10 + 8*q^9 + 7*q^8 + 4*q^7 + 4*q^6 + 2*q^5 + 2*q^4 + q^3 + q^2 + 1
11:
q^110 + q^108 + q^107 + 2*q^106 + 2*q^105 + 4*q^104 + 4*q^103 + 7*q^102 + 8*q^101 + 12*q^100 + 14*q^99 + 20*q^98 + 22*q^97 + 31*q^96 + 36*q^95 + 47*q^94 + 54*q^93 + 70*q^92 + 79*q^91 + 100*q^90 + 113*q^89 + 138*q^88 + 156*q^87 + 188*q^86 + 209*q^85 + 248*q^84 + 276*q^83 + 320*q^82 + 353*q^81 + 406*q^80 + 442*q^79 + 502*q^78 + 544*q^77 + 608*q^76 + 655*q^75 + 725*q^74 + 771*q^73 + 846*q^72 + 895*q^71 + 969*q^70 + 1017*q^69 + 1092*q^68 + 1135*q^67 + 1208*q^66 + 1247*q^65 + 1312*q^64 + 1345*q^63 + 1404*q^62 + 1424*q^61 + 1476*q^60 + 1487*q^59 + 1524*q^58 + 1524*q^57 + 1551*q^56 + 1536*q^55 + 1551*q^54 + 1524*q^53 + 1524*q^52 + 1487*q^51 + 1476*q^50 + 1424*q^49 + 1404*q^48 + 1345*q^47 + 1312*q^46 + 1247*q^45 + 1208*q^44 + 1135*q^43 + 1092*q^42 + 1017*q^41 + 969*q^40 + 895*q^39 + 846*q^38 + 771*q^37 + 725*q^36 + 655*q^35 + 608*q^34 + 544*q^33 + 502*q^32 + 442*q^31 + 406*q^30 + 353*q^29 + 320*q^28 + 276*q^27 + 248*q^26 + 209*q^25 + 188*q^24 + 156*q^23 + 138*q^22 + 113*q^21 + 100*q^20 + 79*q^19 + 70*q^18 + 54*q^17 + 47*q^16 + 36*q^15 + 31*q^14 + 22*q^13 + 20*q^12 + 14*q^11 + 12*q^10 + 8*q^9 + 7*q^8 + 4*q^7 + 4*q^6 + 2*q^5 + 2*q^4 + q^3 + q^2 + 1
Definition
The MacMahon $q$-Catalan polynomial is $\widetilde C_n(q)=q^{\binom n2}C_n(q,q^{-1})$ [4]. It is equivalently the Gaussian binomial quotient in (1).
Parameters
$n$
—   semilength (a nonnegative integer)
Formulas
(1)
$\widetilde C_n(q)=q^{\binom n2}C_n(q,q^{-1})=\frac{1}{[n+1]_q}{2n\brack n}_q$.
(2)
$\widetilde C_n(1)=\frac{1}{n+1}\binom{2n}{n}$.
(3)
$\widetilde C_0(q)=\widetilde C_1(q)=1$ and $\widetilde C_2(q)=q^2+1$.
Programs
(P1)
Sage
from sage.combinat.q_analogues import q_catalan_number

q_catalan_number(12)
References
[1]
J. Haglund, The q,t-Catalan Numbers and the Space of Diagonal Harmonics, University Lecture Series 41, American Mathematical Society, Providence, RI, 2008.
[2]
A. M. Garsia and M. Haiman, A remarkable q,t-Catalan sequence and q-Lagrange inversion, J. Algebraic Combin. 5 (1996), 191-244.
[3]
L. Carlitz and J. Riordan, Two element lattice permutation numbers and their q-generalization, Duke Math. J. 31 (1964), 371-388.
[4]
P. A. MacMahon, Combinatory Analysis, Vol. II, Cambridge University Press, Cambridge, 1916.
Links
Similar tables
$q,t$-Catalan numbers —   stores the two-variable polynomial from which this table is obtained as $q^{\binom n2}C_n(q,q^{-1})$. This table is the likely target for a reader holding the Gaussian-binomial quotient; a reader holding the full dinv-area polynomial wants T262.
Carlitz-Riordan $q$-Catalan numbers —   stores the other standard one-variable $q$-Catalan polynomial, $C_n(q,1)$, while this table stores $q^{\binom n2}C_n(q,q^{-1})$. Neither one-variable table determines the full $C_n(q,t)$.
Data properties
Entries are of type: integral polynomial
How well the digits are known: exact
Table is complete: no (it holds the rows for $3\leq n\leq11$; the omitted rows with $n\leq2$ are the closed forms in (3))
How they were obtained:

Every entry is an exact polynomial with integer coefficients. The generator computes $q^{\binom n2}C_n(q,q^{-1})$ from Sage's qt_catalan_number [5] and checks the result against Sage's q_catalan_number.

more

Before the values were written, the rows were checked against the Gaussian binomial quotient in (1). The checks also verified (2) and the small cases in (3) over the stored range.



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