A217326
Number of self-inverse permutations in S_n with longest increasing subsequence of length 6.
Original entry on oeis.org
1, 6, 41, 209, 1106, 5323, 26069, 122901, 585922, 2747977, 13000269, 61088173, 289186846, 1366147708, 6496681304, 30905464864, 147912712795, 709073550307, 3418258506885, 16517431269189, 80230551304034, 390774361811783, 1912602871119956, 9388456361080840
Offset: 6
a(6) = 1: 123456.
a(7) = 6: 1234576, 1234657, 1235467, 1243567, 1324567, 2134567.
A218266
Number of standard Young tableaux of n cells and height >= 6.
Original entry on oeis.org
1, 7, 49, 273, 1506, 7788, 40161, 202917, 1028170, 5190328, 26375635, 134565795, 692890250, 3596739368, 18877483060, 100131220940, 537718999715, 2922918175965, 16100254700137, 89857257410905, 508473405642250, 2916903963927300, 16969580464205400
Offset: 6
-
b:= proc(n) b(n):= `if`(n<2, 1, b(n-1) +(n-1)*b(n-2)) end:
g:= proc(n) option remember; `if`(n<3, [1, 1, 2][n+1],
((3*n^2+17*n+15)*g(n-1) +(n-1)*(13*n+9)*g(n-2)
-15*(n-1)*(n-2)*g(n-3)) / ((n+4)*(n+6)))
end:
a:= n-> b(n) -g(n):
seq(a(n), n=6..30);
A229068
Number of standard Young tableaux of n cells and height <= 12.
Original entry on oeis.org
1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568503, 2390466, 10349340, 46204720, 211779200, 997134592, 4808141824, 23745792032, 119848119307, 618058083314, 3251373425356, 17442275104496, 95297400355320, 530067682582320, 2998503402985440
Offset: 0
Cf.
A182172,
A001405 (k=2),
A001006 (k=3),
A005817 (k=4),
A049401 (k=5),
A007579 (k=6),
A007578 (k=7),
A007580 (k=8),
A212915 (k=9),
A212916 (k=10),
A229053 (k=11).
-
RecurrenceTable[{-147456 (-5+n) (-4+n) (-3+n) (-2+n) (-1+n) (12+n) a[-6+n]-110592 (-4+n) (-3+n) (-2+n) (-1+n) (29+2 n) a[-5+n]+256 (-3+n) (-2+n) (-1+n) (121272+32786 n+2343 n^2+49 n^3) a[-4+n]+128 (-2+n) (-1+n) (438597+90321 n+5391 n^2+98 n^3) a[-3+n]-16 (-1+n) (8718630+5347213 n+804616 n^2+49754 n^3+1372 n^4+14 n^5) a[-2+n]-8 (27335490+10162354 n+1206473 n^2+63328 n^3+1533 n^4+14 n^5) a[-1+n]+(11+n) (20+n) (27+n) (32+n) (35+n) (36+n) a[n]==0, a[1]==1, a[2]==2, a[3]==4, a[4]==10, a[5]==26, a[6]==76}, a, {n, 20}]
A103140
Number of 3-noncrossing restricted RNA structures with n vertices.
Original entry on oeis.org
1, 1, 1, 2, 5, 14, 40, 119, 364, 1145, 3688, 12139, 40734, 139071, 482214, 1695469, 6036768, 21740969, 79117822, 290674470, 1077306351, 4025068621, 15151115808, 57427176992, 219068962330, 840708048210, 3244438898552, 12586627632549, 49069788882951
Offset: 1
- Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
- Emma Y. Jin, Jing Qin and Christian M. Reidys, Combinatorics of RNA structures with pseudoknots, arXiv:0704.2518 [math.CO], 2007.
- Emma Y. Jin, Jing Qin and Christian M. Reidys, Combinatorics of RNA structures with pseudoknots, Bulletin of Mathematical Biology Vol. 70 (2008) pp. 45-67. See Table 2 on page 62 for details.
-
sf3[n_] := sf3[n] = Sum[Binomial[n, 2 k] (CatalanNumber[k + 2] CatalanNumber[k] - CatalanNumber[k + 1]^2), {k, 0, n/2}]; (* this is A049401 *)
la[0, 0, 0] = 1;
la[?Negative, , ] = la[, ?Negative, ] = la[, , _?Negative] = 0;
la[n_, b1_, b2_] := la[n, b1, b2] = la[n - 2, b1 - 1, b2] + la[n - 1, b1, b2] + la[n - 4, b1, b2 - 2] + la[n - 3, b1, b2 - 1];
a[n_] := Sum[(-1)^(b1 + b2) la[n, b1, b2] sf3[n - 2 (b1 + b2)], {b1, 0, n/2}, {b2, 0, n/2}];
Table[a[n], {n, 30}] (* Andrey Zabolotskiy, Nov 11 2023, from eqs. (4.2), (4.3), and (2.14) by Jin et al. *)
Comments