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A059797 Second in a series of arrays counting standard tableaux by partition type.

%I A059797 #54 Mar 26 2021 19:26:27
%S A059797 2,5,5,9,16,9,14,35,35,14,20,64,90,64,20,27,105,189,189,105,27,35,160,
%T A059797 350,448,350,160,35,44,231,594,924,924,594,231,44,54,320,945,1728,
%U A059797 2100,1728,945,320,54,65,429,1430,3003,4290,4290,3003,1430,429,65
%N A059797 Second in a series of arrays counting standard tableaux by partition type.
%C A059797 The first array in the series is Pascal's triangle, A007318. The initial partition for each subsequent array in the series is chosen as described in A053445. When cells are squared, as in A008459, row sums yield 1, 2, 6, 24, ... (A000142). E.g., (1 + 16 + 36 + 16 + 1) + (25 + 25) = 70 + 50 = 120 using row five from A007318 and row two from this array.
%C A059797 Number of lattice paths from (0,0) to (n,k) in exactly n+k+2 steps. Moves allowed: (0,1), (0,-1), (-1,0) and (1,0). Paths must stay in Quadrant I but may touch the axes. - _Juan Luis Vargas-Molina_, Mar 03 2014
%D A059797 Stanton and White, Constructive Combinatorics, 1986, pp. 84, 91.
%H A059797 Cyann Donnot, Antoine Genitrini, Yassine Herida, <a href="https://hal.sorbonne-universite.fr/hal-02462764">Unranking Combinations Lexicographically: an efficient new strategy compared with others</a>, hal-02462764 [cs] / [cs.DS] / [math] / [math.CO], 2020.
%H A059797 Antoine Genitrini and Martin Pépin, <a href="https://hal.sorbonne-universite.fr/hal-03040740v2">Lexicographic unranking of combinations revisited</a>, hal-03040740v2 [cs.DM] [cs.DS] [math.CO], 2020.
%H A059797 Juan Luis Vargas-Molina, <a href="/A059797/a059797.cpp.txt">C++ Algorithm</a>
%F A059797 T(row, col) = T(row-1, col-1) + T(row-1, col) + A007318(row+2, col+1). - Corrected by _R. J. Mathar_, Jan 27 2011
%F A059797 T(n,k) = (k+1)(n+k+4)FallingFactorial(n+k+2,n)/((n+1)!+n!) n,k >= 0. - _Juan Luis Vargas-Molina_, Mar 03 2014
%e A059797 a(5) = 16 because we can write T(2,2) = T(1,2) + T(1,1) + A007318(4,2) = 5 + 5 + 6.
%e A059797    2;
%e A059797    5,   5;
%e A059797    9,  16,   9;
%e A059797   14,  35,  35,  14;
%e A059797   20,  64,  90,  64,  20;
%e A059797   27, 105, 189, 189, 105,  27;
%e A059797 T(n,k) as a grid for the first few lattice points. Where T(0,0)=2 since there are two ways to get to (0,0) with "0+0+2" steps under the move restrictions.
%e A059797      k = 0  1   2   3    4    5
%e A059797 T(0,k) = 2  5   9   14   20   27
%e A059797 T(1,k) = 5  16  35  64   105  160
%e A059797 T(2,k) = 9  35  90  189  350  594
%e A059797 T(3,k) = 14 64  189 448  924  1728
%e A059797 T(4,k) = 20 105 350 924  2100 4290
%e A059797 T(5,k) = 27 160 594 1728 4290 9504
%e A059797 - _Juan Luis Vargas-Molina_, Mar 03 2014
%p A059797 A059797 := proc(n,k) option remember; if n <0 or k<0 or k>n then 0; else procname(n-1,k-1)+procname(n-1,k)+binomial(n+2,k+1) ; end if; end proc:
%p A059797 seq(seq( A059797(n,k),k=0..n),n=0..15) ; # _R. J. Mathar_, Jan 27 2011
%t A059797 t[n_, k_] /; (n < 0 || k < 0 || k > n) = 0; t[n_, k_] := t[n, k] = t[n - 1, k - 1] + t[n - 1, k] + Binomial[n + 2, k + 1]; Flatten[ Table[ t[n, k], {n, 0, 9}, {k, 0, n}]] (* _Jean-François Alcover_, Dec 20 2011, after _R. J. Mathar_ *)
%t A059797 T[n_, k_] := (k + 1)(n + k +4) FactorialPower[k + n + 2, n]/(n! + (n + 1)!)
%t A059797 MatrixForm[Table[T[n, k], {n, 0, 10}, {k, 0, 10}]] (* _Juan Luis Vargas-Molina_, Mar 03 2014 *)
%Y A059797 Cf. A000041, A000142, A007318, A008459, A053445.
%K A059797 nice,nonn,tabl
%O A059797 0,1
%A A059797 _Alford Arnold_, Feb 22 2001