A217327
Number of self-inverse permutations in S_n with longest increasing subsequence of length 7.
Original entry on oeis.org
1, 7, 55, 319, 1904, 10275, 56135, 294386, 1556323, 8086433, 42298721, 219795160, 1149139210, 5999688692, 31506046052, 165664633982, 875886376212, 4643488263933, 24746018418741, 132328997879066, 711142850556217, 3836134976520394, 20791024498584110
Offset: 7
a(7) = 1: 1234567.
a(8) = 7: 12345687, 12345768, 12346578, 12354678, 12435678, 13245678, 21345678.
A217328
Number of self-inverse permutations in S_n with longest increasing subsequence of length 8.
Original entry on oeis.org
1, 8, 71, 461, 3057, 18225, 109446, 628652, 3628517, 20538209, 116808172, 659078098, 3737763884, 21153403644, 120354760098, 685455514294, 3925104616303, 22535893275064, 130089736567064, 753604985013128, 4388755545268226, 25660332309744370, 150802834643569274
Offset: 8
a(8) = 1: 12345678.
a(9) = 8: 123456798, 123456879, 123457689, 123465789, 123546789, 124356789, 132456789, 213456789.
A218268
Number of standard Young tableaux of n cells and height >= 8.
Original entry on oeis.org
1, 9, 81, 561, 3817, 23881, 147862, 886028, 5288933, 31178901, 183908244, 1081452450, 6381113064, 37719710024, 224141652938, 1337958249446, 8038507929319, 48593807722975, 295913856459150, 1814986751559300, 11220842616565050, 69921225307663290
Offset: 8
-
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<4, [1, 1, 2, 4][n+1],
((4*n^3+78*n^2+424*n+495)*g(n-1) +(n-1)*(34*n^2+280*n
+305)*g(n-2) -2*(n-1)*(n-2)*(38*n+145)*g(n-3) -105*(n-1)
*(n-2)*(n-3)*g(n-4)) / ((n+6)*(n+10)*(n+12)))
end:
a:= n-> b(n) -g(n):
seq(a(n), n=8..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}]
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