A355010 Array read by ascending antidiagonals: T(n, k) is the number of n-core partitions with k corners.

1, 3, 1, 6, 5, 1, 10, 16, 7, 1, 15, 40, 31, 9, 1, 21, 85, 105, 51, 11, 1, 28, 161, 295, 219, 76, 13, 1, 36, 280, 721, 771, 396, 106, 15, 1, 45, 456, 1582, 2331, 1681, 650, 141, 17, 1, 55, 705, 3186, 6244, 6083, 3235, 995, 181, 19, 1, 66, 1045, 5985, 15156, 19348, 13663, 5685, 1445, 226, 21, 1

Offset: 2

Author

Stefano Spezia, Jun 15 2022

Keywords

• nonn
• tabl

Comments

T(n, k) is also equal to the number of cornerless Motzkin paths of length 2*k + n - 1 with n - 1 flat steps (see Theorem 3.3 and Proposition 3.4 at pp. 13 - 14 in Cho et al.).

In proposition 3.4 in Cho et al., the Narayana number is defined as N(k, i) = binomial(k, i)*binomial(k, i-1)/k, unlike A001263.

Examples

			The array begins:
    1,  1,   1,   1,    1,    1,    1,    1, ...
    3,  5,   7,   9,   11,   13,   15,   17, ...
    6, 16,  31,  51,   76,  106,  141,  181, ...
   10, 40, 105, 219,  396,  650,  995, 1445, ...
   15, 85, 295, 771, 1681, 3235, 5685, 9325, ...
   ...
		

Links

• Hyunsoo Cho, JiSun Huh, Hayan Nam, and Jaebum Sohn, Combinatorics on bounded free Motzkin paths and its applications, arXiv:2205.15554 [math.CO], 2022.

Crossrefs

Cf. A000012 (n = 2), A001263, A005408 (n = 3), A005891 (n = 4), A006007, A063490 (n = 5), A160747 (n = 6), A161680 (k = 1), A355011.

Programs

Mathematica

T[n_,k_]:=Sum[Binomial[k,i]Binomial[k,i-1]Binomial[n+2(k-i),2k]/k,{i,Min[k,Floor[n/2]]}]; Flatten[Table[T[n-k+1,k],{n,2,12},{k,1,n-1}]]

Formula

T(n, k) = Sum_{i=1..min(k,floor(n/2))} N(k, i)*binomial(n+2*(k-i), 2*k), where N(k, i) = binomial(k, i)*binomial(k, i-1)/k. (See proposition 3.4 in Cho et al.).

T(n, 2) = A006007(n-1).