A182172
Number A(n,k) of standard Young tableaux of n cells and height <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 4, 6, 1, 0, 1, 1, 2, 4, 9, 10, 1, 0, 1, 1, 2, 4, 10, 21, 20, 1, 0, 1, 1, 2, 4, 10, 25, 51, 35, 1, 0, 1, 1, 2, 4, 10, 26, 70, 127, 70, 1, 0, 1, 1, 2, 4, 10, 26, 75, 196, 323, 126, 1, 0, 1, 1, 2, 4, 10, 26, 76, 225, 588, 835, 252, 1, 0
Offset: 0
A(4,2) = 6, there are 6 standard Young tableaux of 4 cells and height <= 2:
+------+ +------+ +---------+ +---------+ +---------+ +------------+
| 1 3 | | 1 2 | | 1 3 4 | | 1 2 4 | | 1 2 3 | | 1 2 3 4 |
| 2 4 | | 3 4 | | 2 .-----+ | 3 .-----+ | 4 .-----+ +------------+
+------+ +------+ +---+ +---+ +---+
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, 2, 2, 2, ...
0, 1, 3, 4, 4, 4, 4, 4, 4, ...
0, 1, 6, 9, 10, 10, 10, 10, 10, ...
0, 1, 10, 21, 25, 26, 26, 26, 26, ...
0, 1, 20, 51, 70, 75, 76, 76, 76, ...
0, 1, 35, 127, 196, 225, 231, 232, 232, ...
0, 1, 70, 323, 588, 715, 756, 763, 764, ...
Columns k=0-12 give:
A000007,
A000012,
A001405,
A001006,
A005817,
A049401,
A007579,
A007578,
A007580,
A212915,
A212916,
A229053,
A229068.
-
h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
+add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
end:
g:= proc(n, i, l) option remember;
`if`(n=0, h(l), `if`(i<1, 0, `if`(i=1, h([l[], 1$n]),
g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
end:
A:= (n, k)-> g(n, k, []):
seq(seq(A(n, d-n), n=0..d), d=0..15);
-
h[l_List] := Module[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_List] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Array[1&, n]]], g [n, i-1, l] + If[i > n, 0, g[n-i, i, Append[l, i]]]]]];
a[n_, k_] := g[n, k, {}];
Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 06 2013, translated from Maple *)
A240609
Number of n-length words w over a 3-ary alphabet such that w is empty or a prefix z concatenated with letter a_i and i=1 or 0 < #(z,a_{i-1}) >= #(z,a_i), where #(z,a_i) counts the occurrences of the i-th letter in z.
Original entry on oeis.org
1, 1, 2, 5, 13, 35, 94, 254, 688, 1872, 5115, 14038, 38689, 107055, 297336, 828699, 2317098, 6498114, 18273861, 51521238, 145604868, 412407942, 1170507375, 3328570513, 9482518041, 27059673745, 77340925350, 221382318131, 634578781229, 1821388557507
Offset: 0
a(3) = 5: 111, 112, 121, 122, 123.
a(4) = 13: 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1221, 1223, 1231, 1232, 1233.
a(5) = 35: 11111, 11112, 11121, 11122, 11123, 11211, 11212, 11213, 11221, 11222, 11223, 11231, 11232, 11233, 12111, 12112, 12113, 12121, 12122, 12123, 12131, 12132, 12133, 12211, 12212, 12213, 12231, 12233, 12311, 12312, 12313, 12321, 12323, 12331, 12332.
-
a:= proc(n) option remember; `if`(n<3, [1, 1, 2][n+1],
((87*n^5-380*n^4-95*n^3+848*n^2-76*n+96) *a(n-1)
+(n-1)*(29*n^4-117*n^3+228*n^2+404*n-528) *a(n-2)
-3*(n-1)*(n-2)*(29*n^3-59*n^2-34*n-96) *a(n-3))/
((n-2)*(n+4)*(29*n^3-146*n^2+171*n-150)))
end:
seq(a(n), n=0..35);
-
b[n_, k_, l_] := b[n, k, l] = If[n == 0, 1, If[Length[l] < k, b[n - 1, k, Append[l, 1]], 0] + Sum[If[i == 1 || l[[i]] <= l[[i - 1]], b[n - 1, k, ReplacePart[l, i -> l[[i]] + 1]], 0], {i, 1, Length[l]}]];
a[n_] := b[n, Min[3, n], {}];
a /@ Range[0, 35] (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz in A240608 *)
A240610
Number of n-length words w over a 4-ary alphabet such that w is empty or a prefix z concatenated with letter a_i and i=1 or 0 < #(z,a_{i-1}) >= #(z,a_i), where #(z,a_i) counts the occurrences of the i-th letter in z.
Original entry on oeis.org
1, 1, 2, 5, 14, 45, 149, 509, 1756, 6082, 21246, 74125, 260940, 917060, 3251573, 11514563, 41115372, 146687475, 527318119, 1894548198, 6853413645, 24782205276, 90165044683, 327958242274, 1199475359682, 4386106532923, 16118279172044, 59223681674225
Offset: 0
A240611
Number of n-length words w over a 5-ary alphabet such that w is empty or a prefix z concatenated with letter a_i and i=1 or 0 < #(z,a_{i-1}) >= #(z,a_i), where #(z,a_i) counts the occurrences of the i-th letter in z.
Original entry on oeis.org
1, 1, 2, 5, 14, 46, 164, 629, 2511, 10284, 42765, 179411, 757518, 3212026, 13672821, 58390509, 250183486, 1075280768, 4636221163, 20051763960, 86995226538, 378588153837, 1652550862097, 7234850625649, 31766225617819, 139871594177849, 617573798910950
Offset: 0
A240612
Number of n-length words w over a 6-ary alphabet such that w is empty or a prefix z concatenated with letter a_i and i=1 or 0 < #(z,a_{i-1}) >= #(z,a_i), where #(z,a_i) counts the occurrences of the i-th letter in z.
Original entry on oeis.org
1, 1, 2, 5, 14, 46, 165, 650, 2742, 12195, 56388, 267382, 1291017, 6304776, 31043333, 153739806, 764663187, 3817811679, 19117378411, 96034890298, 483633087406, 2442935438558, 12369440072730, 62816259353597, 319763315261222, 1632483727134416, 8354058837330795
Offset: 0
A240613
Number of n-length words w over a 7-ary alphabet such that w is empty or a prefix z concatenated with letter a_i and i=1 or 0 < #(z,a_{i-1}) >= #(z,a_i), where #(z,a_i) counts the occurrences of the i-th letter in z.
Original entry on oeis.org
1, 1, 2, 5, 14, 46, 165, 651, 2770, 12601, 60686, 305658, 1595454, 8547278, 46685641, 258530476, 1446098060, 8148181066, 46165702543, 262699285827, 1500141404881, 8592741193076, 49353003881425, 284185414898313, 1640366631269049, 9490895165369217
Offset: 0
A240614
Number of n-length words w over an 8-ary alphabet such that w is empty or a prefix z concatenated with letter a_i and i=1 or 0 < #(z,a_{i-1}) >= #(z,a_i), where #(z,a_i) counts the occurrences of the i-th letter in z.
Original entry on oeis.org
1, 1, 2, 5, 14, 46, 165, 651, 2771, 12637, 61352, 314472, 1691635, 9472308, 54830409, 325818836, 1975978117, 12171736767, 75862916184, 477079866910, 3020725897374, 19228274412487, 122916363687789, 788480233155175, 5073010198556052, 32724394668837491
Offset: 0
A240615
Number of n-length words w over a 9-ary alphabet such that w is empty or a prefix z concatenated with letter a_i and i=1 or 0 < #(z,a_{i-1}) >= #(z,a_i), where #(z,a_i) counts the occurrences of the i-th letter in z.
Original entry on oeis.org
1, 1, 2, 5, 14, 46, 165, 651, 2771, 12638, 61397, 315507, 1708420, 9693168, 57360909, 352099643, 2229930365, 14494768448, 96229497555, 649761232948, 4446218110646, 30745322129155, 214353097904732, 1504152530715607, 10609439744527674, 75146372454242527
Offset: 0
A240616
Number of n-length words w over a 10-ary alphabet such that w is empty or a prefix z concatenated with letter a_i and i=1 or 0 < #(z,a_{i-1}) >= #(z,a_i), where #(z,a_i) counts the occurrences of the i-th letter in z.
Original entry on oeis.org
1, 1, 2, 5, 14, 46, 165, 651, 2771, 12638, 61398, 315562, 1709960, 9723253, 57831688, 358445901, 2306799588, 15354546420, 105275618325, 740481162174, 5321875886529, 38940082552276, 289132258826130, 2172495189822333, 16480880960094288, 125993420982428718
Offset: 0
A240617
Number of n-length words w over an n-ary alphabet such that w is empty or a prefix z concatenated with letter a_i and i=1 or 0 < #(z,a_{i-1}) >= #(z,a_i), where #(z,a_i) counts the occurrences of the i-th letter in z.
Original entry on oeis.org
1, 1, 2, 5, 14, 46, 165, 651, 2771, 12638, 61398, 315563, 1710027, 9725543, 57886132, 359476571, 2323518017, 15597303757, 108522502933, 781283103284, 5810573402452, 44579551962460, 352360829928066, 2865876353091752, 23958937424819348, 205673285003637629
Offset: 0
a(0) = 1: the empty word.
a(1) = 1: 1.
a(2) = 2: 11, 12.
a(3) = 5: 111, 112, 121, 122, 123.
a(4) = 14: 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1221, 1223, 1231, 1232, 1233, 1234.
a(5) = 46: 11111, 11112, 11121, ..., 12343, 12344, 12345.
a(6) = 165: 111111, 111112, 111121, ..., 123454, 123455, 123456.
Showing 1-10 of 10 results.
Comments