login
Number T(n,k) of distinct values of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n with largest part = k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
3

%I #30 Aug 26 2019 17:51:34

%S 1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,3,3,2,1,1,1,3,4,3,2,1,1,1,4,5,5,3,2,

%T 1,1,1,4,7,6,5,3,2,1,1,1,5,8,9,7,5,3,2,1,1,1,5,10,10,10,7,5,3,2,1,1,1,

%U 6,12,14,12,11,7,5,3,2,1,1,1,6,14,16,17,13,11,7,5,3,2,1,1

%N Number T(n,k) of distinct values of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n with largest part = k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%C Differs from A008284 first at T(11,4).

%H Alois P. Heinz, <a href="/A215521/b215521.txt">Rows n = 1..100, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>

%e T(4,2) = 2 = |{4!/(2!*2!), 4!/(2!*1!*1!)}| = |{6, 12}|.

%e T(7,4) = 3 = |{35, 105, 210}|.

%e T(8,3) = 5 = |{560, 1120, 1680, 3360, 6720}|.

%e T(11,4) = 10 = |{11550, 34650, 46200, 69300, 138600, 207900, 277200, 415800, 831600, 1663200}|.

%e Triangle T(n,k) begins:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 2, 1, 1;

%e 1, 2, 2, 1, 1;

%e 1, 3, 3, 2, 1, 1;

%e 1, 3, 4, 3, 2, 1, 1;

%e 1, 4, 5, 5, 3, 2, 1, 1;

%e 1, 4, 7, 6, 5, 3, 2, 1, 1;

%e 1, 5, 8, 9, 7, 5, 3, 2, 1, 1;

%e 1, 5, 10, 10, 10, 7, 5, 3, 2, 1, 1;

%e ...

%p b:= proc(n, i) option remember; `if`(n=0, {1}, `if`(i<1, {},

%p {b(n, i-1)[], seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=1..n/i)}))

%p end:

%p T:= (n, k)-> nops(b(n-k, min(k, n-k))):

%p seq(seq(T(n, k), k=1..n), n=1..15);

%t b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, Join[b[n, i - 1], Table[ b[n - i*j, i - 1] *i!^j, {j, 1, n/i}] // Flatten]] // Union]; T[n_, k_] := Length[b[n, k]]; Table[Table[T[n - k, Min[k, n - k]], {k, 1, n}], {n, 1, 15}] // Flatten (* _Jean-François Alcover_, Jan 21 2015, after _Alois P. Heinz_ *)

%Y Columns k=1-3 give: A000012 (for n>0), A004526, A069905(n) = A001399(n-3) for n>=3.

%Y T(2*n,n) gives: A070289.

%Y Cf. A008284, A215520.

%K nonn,tabl

%O 1,8

%A _Alois P. Heinz_, Aug 14 2012