A289713 -id:A289713 - cp's OEIS frontend

A289710 Triangle T(n,r) read by rows: order of the semigroup of orientation-preserving partial transformations of n elements with breath r.

1, 1, 1, 1, 4, 4, 1, 9, 27, 24, 1, 16, 96, 208, 128, 1, 25, 250, 950, 1325, 610, 1, 36, 540, 3120, 7290, 7416, 2742, 1, 49, 1029, 8330, 28665, 47922, 38563, 11970, 1, 64, 1792, 19264, 90720, 219968, 287168, 191808, 51424, 1, 81, 2916, 40068, 246078, 806274, 1509732, 1619676, 926073, 218718

Offset: 0

R. J. Mathar, Sep 02 2017

			1 ;
1 1;
1 4 4;
1 9 27 24;
1 16 96 208 128;
1 25 250 950 1325 610;
1 36 540 3120 7290 7416 2742;
1 49 1029 8330 28665 47922 38563 11970;
1 64 1792 19264 90720 219968 287168 191808 51424;
1 81 2916 40068 246078 806274 1509732 1619676 926073 218718;

A. Umar, Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations, J. Int. Seq. 14 (2011) # 11.7.5, Corollary 9, Table 2.1

Cf. A092364 (column r=2), A289713 (row sums)

A289710 := proc(n,r)
    if r = 0 then
        1;
    else
        r*binomial(n,r)*binomial(n+r-1,n-1)-n*(r-1)*binomial(n,r) ;
    end if ;
end proc:

A289711 Triangle T(n,p) read by rows: the order of the semigroup of orientation-preserving partial transformations of n elements with height p.

Original entry on oeis.org

1, 1, 1, 1, 6, 2, 1, 21, 36, 3, 1, 60, 288, 96, 4, 1, 155, 1600, 1200, 200, 5, 1, 378, 7200, 9600, 3600, 360, 6, 1, 889, 28224, 58800, 39200, 8820, 588, 7, 1, 2040, 100352, 301056, 313600, 125440, 18816, 896, 8, 1, 4599, 331776, 1354752, 2032128, 1270080, 338688

Offset: 0

R. J. Mathar, Sep 02 2017

			1 ;
1 1 ;
1 6 2 ;
1 21 36 3 ;
1 60 288 96 4;
1 155 1600 1200 200 5;
1 378 7200 9600 3600 360 6;
1 889 28224 58800 39200 8820 588 7;
1 2040 100352 301056 313600 125440 18816 896 8 ;
1 4599 331776 1354752 2032128 1270080 338688 36288 1296 9;
1 10230 1036800 5529600 11289600 10160640 4233600 806400 64800 1800 10 ;

A. Umar, Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations, J. Int. Seq. 14 (2011) # 11.7.5, Corollary 10, Table 2.2

Cf. A066524 (column p=1), A289713 (row sums).

A289711 := proc(n,p)
    if p = 0 then
        1;
    elif p = 1 then
        n*(2^n-1) ;
    else
        p*2^(n-p)*binomial(n,p)^2 ;
    end if ;
end proc:

A289714 Triangle T(n,k) read by rows: the number of semigroups of orientation-preserving partial transformations on n element with right waist k.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 1, 7, 19, 34, 1, 15, 63, 135, 235, 1, 31, 191, 471, 911, 1556, 1, 63, 543, 1503, 3183, 5883, 9969, 1, 127, 1471, 4495, 10319, 20483, 37031, 62602, 1, 255, 3839, 12799, 31615, 67007, 128607, 229743, 388343, 1, 511, 9727, 35071, 92671, 208735, 423583, 796687, 1412863, 2389768, 1

Offset: 0

R. J. Mathar, Sep 02 2017

			1 ;
1 1 ;
1 3 5 ;
1 7 19 34 ;
1 15 63 135 235;
1 31 191 471 911 1556 ;
1 63 543 1503 3183 5883 9969 ;
1 127 1471 4495 10319 20483 37031 62602 ;
1 255 3839 12799 31615 67007 128607 229743 388343 ;
1 511 9727 35071 92671 208735 423583 796687 1412863 2389768 ;
1 1023 24063 93183 262143 625023 1336383 2638143 4894623 8637363 14621533 ;

A. Umar, Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations, J. Int. Seq. 14 (2011) # 11.7.5

Cf. A289713 (row sums), A000225 (column 1), A053545 (column 2)

A289714 := proc(n,k)
    if k = 0 then
        1;
    else
        n*add(binomial(n-1,r-1)*binomial(r+k-2,r-1),r=1..n)-(n-2)*2^(n-1)-1 ;
    end if ;
end proc:

cp's OEIS Frontend

A289710 Triangle T(n,r) read by rows: order of the semigroup of orientation-preserving partial transformations of n elements with breath r.

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A289711 Triangle T(n,p) read by rows: the order of the semigroup of orientation-preserving partial transformations of n elements with height p.

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A289714 Triangle T(n,k) read by rows: the number of semigroups of orientation-preserving partial transformations on n element with right waist k.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple