A338135 Irregular triangle read by rows: Row p gives number of non-overlapping clusters of 2q-plets joining 2p points on a circle, i.e., number of noncrossing partitions from A134264 with h_k for k odd replaced by zero.

1, 1, 2, 1, 6, 5, 1, 8, 4, 28, 14, 1, 10, 10, 45, 45, 120, 42, 1, 12, 12, 6, 66, 132, 22, 220, 330, 495, 132, 1, 14, 14, 14, 91, 182, 91, 91, 364, 1092, 364, 1001, 2002, 2002, 429

Offset: 1

Author

Tom Copeland, Oct 11 2020

Keywords

• nonn
• easy
• tabf

Comments

This combinatorial problem arises in relating connected and disconnected Green functions associated to a "zero-dimensional" quantum field theory presented by Brezin et al. in "Planar Diagrams" via Eqn. 31 on p. 42.

Appears to be a refinement of A120986 and A108767 in that summing the coefficients of partitions with the same sum of exponents gives the rows or reverse rows of the two entries; for example, row 4 here becomes x + 8 xx + 4 x^2 + 28 x^2x + 14 x^4 = x + 12 x^2 + 28 x^3 + 14 x^4, which is row 4 of A108767. In short, replace each g_k or (k) by x in the formula section here to obtain the coarser entry or its reverse from this refined entry, apparently.

This also gives the relationship between moments and free cumulants in free probability theory restricted to an even number of noncrossing partitions as given by restricting the similar enumeration formuia on p. 34 of Novak and LaCroix to b_{2n} = G_{2n} and K_{2q} = g_{2q}. This is consistent with setting h_k to zero for odd k in A134264, e.g., doing so for the coefficients of t^7 for g(t) there gives G_6 here.

A125181 is another version of A134264, providing interpretations in terms of Dyck paths and trees.

Examples

			row 1: G_2  = g_2
row 2: G_4  = g_4  +  2 g_2^2
row 3: G_6  = g_6  +  6 g_2 g_4 +  5 g_2^3
row 4: G_8  = g_8  +  8 g_2 g_6 +  4 g_4^2   +  28 g_2^2 g_4 + 14 g_2^4
row 5: G_10 = g_10 + 10 g_2 g_8 + 10 g_4 g_6 +  45 g_2^2 g_6 + 45 g_2 g_4^2
              + 120 g_2^3 g_4  + 42 g_2^5
_____________
In the notation of Abramowitz and Stegun p. 831 with indices of the partitions above divided by 2;
R_1  = (1)
R_2  = (2)  +  2 (1)^2
R_3  = (3)  +  6 (1) (2) +  5 (1)^3
R_4  = (4)  +  8 (1) (3) +  4 (2)^2   +  28 (1)^2 (2) + 14 (1)^4
R_5  = (5)  + 10 (1) (4) + 10 (2) (3) +  45 (1)^2 (3) + 45 (1) (2)^2
        + 120 (1)^3 (2) + 42 (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].
• E. Brezin, C. Itzykson, G. Parisi, and J. B. Zuber, Planar Diagrams, Comm. Math. Phys., 59, p. 35-51, Springer-Verlag, 1978, (link via Project Euclid).
• Freddy Cachazo and Bruno Giménez Umbert, Connecting Scalar Amplitudes using The Positive Tropical Grassmannian, arXiv:2205.02722 [hep-th], 2022.
• J. Novak and M. LaCroix, Three lectures on free probability, arXiv:1205.2097 [math.CO], 2012.

Crossrefs

Cf. A000108, A120986, A108767, A125181, A134264.

Programs

Mathematica

Table[(2 n)!/((2 n + 1 - Length@p)! Product[r!, {r, Last /@ Tally[p]}]), {n, 5}, {p, Sort[Sort /@ IntegerPartitions[n]]}] // Flatten (* Andrey Zabolotskiy, Mar 07 2024 *)

Formula

Under the constraint 2p = Sum_{q} 2q r_q, then G_{2p} = Sum_{r_q >= 0} [(2p)! / (2p + 1 - Sum_{q} r_q)! ] (g_2^r_1 /(r_1)!) (g_4^r_2 / (r_2)!) ... (g_{2q}^r_q / (r_q)!) where g_{2k} are the connected Green functions.

With R_p = G_{2p} and N_q = g_{2q}, then R_p = Sum_{r_q >= 0} [(2p)! / (2p + 1 - Sum_{q} r_q)! ] (N_1^r_1 /(r_1)!) (N_2^r_2 / (r_2)!) ... (N_{q}^r_q / (r_q)!) where N_q are the partitions in Abramowitz and Stegun on p. 831.

Coefficients of the final terms g_{2}^p = (1)^p are the Catalan numbers A000108.

Extensions

Rows 6-7 from Andrey Zabolotskiy, Mar 07 2024