A278987 Array read by antidiagonals downwards: T(b,n) = number of words of length n over an alphabet of size b that are in standard order and which have the property that every letter that appears in the word is repeated.

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 1, 1, 0, 1, 11, 4, 1, 1, 0, 1, 26, 11, 4, 1, 1, 0, 1, 57, 41, 11, 4, 1, 1, 0, 1, 120, 162, 41, 11, 4, 1, 1, 0, 1, 247, 610, 162, 41, 11, 4, 1, 1, 0, 1, 502, 2165, 715, 162, 41, 11, 4, 1, 1, 0, 1, 1013, 7327, 3425, 715, 162, 41, 11, 4, 1, 1, 0

Offset: 1

Author

Joerg Arndt and N. J. A. Sloane, Dec 06 2016

Keywords

• nonn
• tabl

Comments

We study words made of letters from an alphabet of size b, where b >= 1. We assume the letters are labeled {1,2,3,...,b}. There are b^n possible words of length n.

We say that a word is in "standard order" if it has the property that whenever a letter i appears, the letter i-1 has already appeared in the word. This implies that all words begin with the letter 1.

Examples

			The array begins:
0,.1,..1,...1,...1,...1,...1,....1..; b=1,
0,.1,..4,...8,..16,..32,..64,..128..; b=2,
0,.1,..4,..14,..41,.122,.365,.1094..; b=3,
0,.1,..4,..14,..51,.187,.715,.2795..; b=4,
0,.1,..4,..14,..51,.202,.855,.3845..; b=5,
0,.1,..4,..14,..51,.202,.876,.4111..; b=6,
...
Rows b=1 through b=4 of the array are A000012, A000295 (or A130103), A278988, A278989.
		

Links

• Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"

Crossrefs

The words for b=9 are listed in A273978.

Programs

Maple

with(combinat);
A008299 := proc(n,k) local i,j,t1;
if k<1 or k>floor(n/2) then t1:=0; else
t1 := add( (-1)^i*binomial(n, i)*add( (-1)^j*(k - i - j)^(n - i)/(j!*(k - i - j)!), j = 0..k - i), i = 0..k); fi; t1; end;
f3:=proc(L,b) global A008299; local i; add(A008299(L,i),i=1..b); end;
Q3:=b->[seq(f3(L,b),L=1..40)];
for b from 1 to 6 do lprint(Q3(b)); od:

Formula

The number of words of length n over an alphabet of size b that are in standard order and in which every symbol that appears in a word is repeated is Sum_{j = 1..b} A008299(n,j).