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 A007578 Number of Young tableaux of height <= 7. (Formerly M1219) 15
 1, 1, 2, 4, 10, 26, 76, 232, 763, 2611, 9415, 35135, 136335, 544623, 2242618, 9463508, 40917803, 180620411, 813405580, 3728248990, 17377551032, 82232982872, 394742985974, 1919885633178, 9453682648281, 47086636037601, 237071351741426, 1205689994416252 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also the number of n-length words w over 7-ary alphabet {a1,a2,...,a7} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a7), where #(z,x) counts the letters x in word z. - Alois P. Heinz, May 30 2012 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 Andrei Asinowski, Axel Bacher, Cyril Banderier, 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). F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Preprint. (Annotated scanned copy) F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468. Juan B. Gil, Peter R. W. McNamara, Jordan O. Tirrell, Michael D. Weiner, From Dyck paths to standard Young tableaux, arXiv:1708.00513 [math.CO], 2017. Index entries for sequences related to Young tableaux. FORMULA a(n) ~ 45/32 * 7^(n+21/2)/(Pi^(3/2)*n^(21/2)). - Vaclav Kotesovec, Sep 11 2013 MAPLE 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, h([l[], 1\$n]), `if`(i<1, 0, g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i]))))) end: a:= n-> g(n, 7, []): seq(a(n), n=0..30); # Alois P. Heinz, Apr 10 2012 # second Maple program a:= proc(n) option remember; `if`(n<4, [1, 1, 2, 4][n+1], ((4*n^3+78*n^2+424*n+495)*a(n-1) +(n-1)*(34*n^2+280*n+305)*a(n-2) -2*(n-1)*(n-2)*(38*n+145)*a(n-3) -105*(n-1)*(n-2)*(n-3)*a(n-4)) / ((n+6)*(n+10)*(n+12))) end: seq(a(n), n=0..30); # Alois P. Heinz, Oct 12 2012 MATHEMATICA RecurrenceTable[{105 (-3+n) (-2+n) (-1+n) a[-4+n]+2 (-2+n) (-1+n) (145+38 n) a[-3+n]-(-1+n) (305+280 n+34 n^2) a[-2+n]+(-495-424 n-78 n^2-4 n^3) a[-1+n]+(6+n) (10+n) (12+n) a[n]==0, a[1]==1, a[2]==2, a[3]==4, a[4]==10}, a, {n, 20}] (* Vaclav Kotesovec, Sep 11 2013 *) CROSSREFS Column k=7 of A182172. - Alois P. Heinz, May 30 2012 Sequence in context: A303930 A007123 A220871 * A239079 A007580 A239080 Adjacent sequences: A007575 A007576 A007577 * A007579 A007580 A007581 KEYWORD nonn AUTHOR Simon Plouffe EXTENSIONS More terms from Alois P. Heinz, Apr 10 2012 STATUS approved

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Last modified September 17 14:23 EDT 2024. Contains 375987 sequences. (Running on oeis4.)