A392945 - OEIS

%I #23 Jan 30 2026 21:15:54

%S 1,1,1,2,1,3,3,6,1,4,6,4,12,12,12,24,1,5,10,10,5,20,30,20,30,30,20,60,

%T 60,60,60,120,1,6,15,20,15,6,30,60,60,30,60,90,60,60,60,30,120,180,

%U 120,180,180,120,180,180,180,120,360,360,360,360,360,720

%N Irregular triangular array T(n,k) read by rows. Row n gives the number of permutations of {1,2,...,n} whose descent set is contained in S for each subset S of {1,2,...,n-1} where the subsets are ordered first by size and then lexicographically (Cf. A082185).

%C Let S be the k-th subset of [n-1]. T(n,k) is the number of maximal chains in the S-rank selected poset of the Boolean lattice on [n]. See Stanley reference below.

%D Richard Stanley, Enumerative Combinatorics, Volume 1, Second Edition, Section 3.13.

%H Alois P. Heinz, <a href="/A392945/b392945.txt">Rows n = 0..15, flattened</a>

%F Let S = {s_1<s_2<...<s_k}. T(n,S) = multinomial(n;s_1,s_2-s_1,s_3-s_2,...,n-s_k). See Stanley reference above and Example 2.2.4.

%F T(n,S) = Sum_A335845(n,T) where the sum is over every subset T of S.

%e Triangle begins:

%e   1;

%e   1;

%e   1, 2;

%e   1, 3,  3,  6;

%e   1, 4,  6,  4, 12, 12, 12, 24;

%e   1, 5, 10, 10,  5, 20, 30, 20, 30, 30, 20, 60, 60, 60, 60, 120;

%e   ...

%p T:= proc(n) option remember; uses combinat; `if`(n=0, 1, map(l->

%p      `if`(l=[], 1, multinomial(n, l[1], l[i]-l[i-1]$i=2..nops(l),

%p       n-l[-1])), [seq(choose([$1..n-1], i)[], i=0..n-1)])[])

%p     end:

%p seq(T(n), n=0..6);  # _Alois P. Heinz_, Jan 27 2026

%t descentset[permutation_] := Flatten[Position[Sign[Differences[permutation]], -1]]; alpha[n_, subset_] := Select[Permutations[Range[n]],MemberQ[Subsets[subset], descentset[#]] &] // Length; flagfvectorboolean[n_] := Map[alpha[n, #] &,Subsets[Range[n - 1]]]; Table[flagfvectorboolean[n], {n, 0, 6}]

%Y Cf. A000142, A335845, A082185, A000670 (row sums).

%K nonn,look,tabf

%O 0,4

%A _Geoffrey Critzer_, Jan 27 2026