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A373949 Triangle read by rows where T(n,k) is the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) yields a composition of k.

%I A373949 #15 Mar 21 2025 10:12:58
%S A373949 1,0,1,0,1,1,0,1,0,3,0,1,1,2,4,0,1,0,4,4,7,0,1,1,5,6,5,14,0,1,0,6,10,
%T A373949 10,14,23,0,1,1,6,14,12,29,26,39,0,1,0,9,16,19,40,54,46,71,0,1,1,8,22,
%U A373949 22,64,82,96,92,124,0,1,0,10,26,30,82,137,144,204,176,214
%N A373949 Triangle read by rows where T(n,k) is the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) yields a composition of k.
%H A373949 John Tyler Rascoe, <a href="/A373949/b373949.txt">Rows n = 0..130, flattened</a>
%F A373949 G.f.: 1/(1 - Sum_{i>0} (y^i * x^i)/(1 + x^i * (y^i - 1))). - _John Tyler Rascoe_, Mar 20 2025
%e A373949 Triangle begins:
%e A373949    1
%e A373949    0   1
%e A373949    0   1   1
%e A373949    0   1   0   3
%e A373949    0   1   1   2   4
%e A373949    0   1   0   4   4   7
%e A373949    0   1   1   5   6   5  14
%e A373949    0   1   0   6  10  10  14  23
%e A373949    0   1   1   6  14  12  29  26  39
%e A373949    0   1   0   9  16  19  40  54  46  71
%e A373949    0   1   1   8  22  22  64  82  96  92 124
%e A373949    0   1   0  10  26  30  82 137 144 204 176 214
%e A373949    0   1   1  11  32  31 121 186 240 331 393 323 378
%e A373949 Row n = 6 counts the following compositions:
%e A373949   .  (111111)  (222)  (33)     (3111)   (411)   (6)
%e A373949                       (2211)   (1113)   (114)   (51)
%e A373949                       (1122)   (1221)   (1311)  (15)
%e A373949                       (21111)  (12111)  (1131)  (42)
%e A373949                       (11112)  (11211)  (2112)  (24)
%e A373949                                (11121)          (141)
%e A373949                                                 (321)
%e A373949                                                 (312)
%e A373949                                                 (231)
%e A373949                                                 (213)
%e A373949                                                 (132)
%e A373949                                                 (123)
%e A373949                                                 (2121)
%e A373949                                                 (1212)
%e A373949 For example, the composition (1,2,2,1) with compression (1,2,1) is counted under T(6,4).
%t A373949 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#]]==k&]], {n,0,10},{k,0,n}]
%o A373949 (PARI)
%o A373949 T_xy(row_max) = {my(N=row_max+1, x='x+O('x^N), h=1/(1-sum(i=1,N, (y^i*x^i)/(1+x^i*(y^i-1))))); vector(N, n, Vecrev(polcoeff(h, n-1)))}
%o A373949 T_xy(13) \\ _John Tyler Rascoe_, Mar 20 2025
%Y A373949 Column k = n is A003242 (anti-runs or compressed compositions).
%Y A373949 Row-sums are A011782.
%Y A373949 Same as A373951 with rows reversed.
%Y A373949 Column k = 3 is A373952.
%Y A373949 This statistic is represented by A373953, difference A373954.
%Y A373949 A114901 counts compositions with no isolated parts.
%Y A373949 A116861 counts partitions by compressed sum, by compressed length A116608.
%Y A373949 A124767 counts runs in standard compositions, anti-runs A333381.
%Y A373949 A240085 counts compositions with no unique parts.
%Y A373949 A333755 counts compositions by compressed length.
%Y A373949 A373948 represents the run-compression transformation.
%Y A373949 Cf. A037201 (halved A373947), A106356, A124762, A238130, A238279, A238343, A285981, A333213, A333489, A373950.
%K A373949 nonn,tabl
%O A373949 0,10
%A A373949 _Gus Wiseman_, Jun 28 2024