A354622 Irregular triangle read by rows: Refined 3-Narayana triangle. Coefficients of partition polynomials of A134264, a refined Narayana triangle enumerating noncrossing partitions, with all h_k other than h_0, h_3, h_6, ..., h_(3n), ... replaced by zero.
1, 1, 3, 1, 9, 12, 1, 12, 6, 66, 55, 1, 15, 15, 105, 105, 455, 273, 1, 18, 18, 9, 153, 306, 51, 816, 1224, 3060, 1428, 1, 21, 21, 21, 210, 420, 210, 210, 1330, 3990, 1330, 5985, 11970, 20349, 7752, 1, 24, 24, 24, 12, 276, 552, 552, 276, 276, 2024, 6072, 3036, 6072, 506, 10626, 42504, 21252, 42504, 106260, 134596, 43263
Offset: 1
Examples
Triangle begins:
1;
1, 3;
1, 9, 12;
1, 12, 6, 66, 55;
1, 15, 15, 105, 105, 455, 273;
...
Row 1: G_3 = g_3
row 2: G_6 = g_6 + 3 g_3^2
row 3: G_9 = g_9 + 9 g_3 g_6 + 12 g_3^3
row 4: G_12 = g_12 + 12 g_3 g_9 + 6 g_6^2 + 66 g_3^2 g_6 + 55 g_3^4
row 5: G_15 = g_15 + 15 g_3 g_12 + 15 g_6 g_9 + 105 g_3^2 g_9 + 105 g_3 g_6^2
+ 455 g_3^3 g_6 + 273 g_3^5.
.
In the notation of Abramowitz and Stegun p. 831 with indices of the partitions above divided by 3 and partition indeterminates h_n denoted (n):
R_1 = (1);
R_2 = (2) + 3 (1)^2;
R_3 = (3) + 9 (1) (2) + 12 (1)^3;
R_4 = (4) + 12 (1) (3) + 6 (2)^2 + 66 (1)^2 (2) + 55 (1)^4;
R_5 = (5) + 15 (1) (4) + 15 (2) (3) + 105 (1)^2 (3) + 105 (1) (2)^2 + 455 (1)^3(2)
+ 273 (1)^5.
Links
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]
- F. Cachazo and B. Umbert, Connecting Scalar Amplitudes using The Positive Tropical Grassmannian, arXiv preprint arXiv:2205.02722 [hep-th], 2022.
- MathOverflow, Combinatorics of iterated composition of noncrossing partition polynomials, a question posed by Tom Copeland, 2022.
- Eric Weisstein's World of Mathematics, Dyck Path.
Crossrefs
Programs
-
Mathematica
Table[Binomial[Total[y], Length[y]-1] (Length[y]-1)! / Product[Count[y, i]!, {i, Max@@y}], {n, 8}, {y, Sort[Sort /@ IntegerPartitions[3n, n, Range[3, 3n, 3]]]}] // Flatten (* Andrey Zabolotskiy, Feb 19 2024, using Gus Wiseman's code for A134264 *) -
PARI
\\ Compare with A134264 C(v)={my(n=vecsum(v), S=Set(v)); n!/((n-#v+1)!*prod(i=1, #S, my(x=S[i]); (#select(y->y==x, v))!))} row(n)=[C(3*Vec(p)) | p<-partitions(n)] { for(n=1, 7, print(row(n))) } \\ Andrew Howroyd, Feb 19 2024
Formula
Coefficients of the monomials are those of the surviving monomials of the partition polynomials of A134264 after zeroing all indeterminates other than h_0, h_3, h_6, h_9, ..., h_(3n), .... The multinomial coefficients of A125181 also apply for G_n, giving the coefficient of the monomial h_1^e_1 h_2^e_2 ... h_n^n of R_n with se := e_1 + e_2 + ... + e_n as (3n)! / ((3n-se+1)! (e_1)! (e_2)! ... (e_n)!).
1*e_1 + 2*e_2 + ... + n*e_n = n for each monomial of R_n.
The partition polynomials R_n = N_n^3 of this entry can be determined from those of A338135, N_n^2, by substituting the partition polynomials of A134264, N_n, for the indeterminate h_n = (n) of N_n^2 or by doing the same for A134264 twice. E.g., N_1(h_1) = h_1, N_2(h_1,h_2) = h_2 + h_1^2, so N_2^2(h_1,h_2) = N_2(N_1,N_2) = N_2 + N_1 = h_2 + h_1^2 + h_1^2 = h_2 + 2 h_1^2 and N_2^3(h_1,h_2) = N_2^2(N_1,N_2) = N_2 + 2 N_1^2 = h_2 + h_1^2 + 2 h_1^2 = h_2 + 3 h_1^2.
Reduces with all indeterminates h_n = (n) = t to A173020.
The coefficient of the monomial h_1^n is (3*n)! / ((3*n-n+1)! n!) = A001764(n) (see also A179848 and A235534). In general, the coefficients of these monomials of the refined (m+1)-Narayana polynomials are the Fuss-Catalan sequence associated to the row sums of the refined m-Narayana polynomials.
The coefficient of the monomial h_1^(n-2) h_2 is (3n)! / ((3n-n+2)! (n-2)!) = A003408(n-2) for n > 1.
The coefficient of the monomial h_1^(n-3) h_3 is (3n)! / ((3n-n+3)! (n-3)!) = A004321(n) for n > 2.
Extensions
Rows 6-8 added by Andrey Zabolotskiy, Feb 19 2024
Comments